UFDC Home  Search all Groups  UF Institutional Repository  UF Institutional Repository  UF Theses & Dissertations  Vendor Digitized Files   Help 
Material Information
Subjects
Notes
Record Information

Full Text 
SOME PROBLEMS IN APPROXIMATION THEORY By BRANDON UNDERHILL A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1996 This dissertation is dedicated to the memories of Professor Arun Kumar Varma and Robert Patrick McGrath ACKNOWLEDGEMENTS I express my deep gratitude to the late Professor Arun Kumar Varma for all of his encouragement and many hours of enthusiastic assistance and instruction in the area of Approximation Theory which made this dissertation possible. I extend special thanks to Professor J. Szabados for his kind assistance on a trip to Gainesville and thereafter. Similarly, special thanks go out to Professor A. Sharma and Profes sor T.M. Mills for their suggestions and encouragement. I thank the members of my supervisory committee for their helpful comments and suggestions, and I thank Mrs. Arlene Williams for her expert assistance. I thank my friends for keeping me distracted, and finally, I thank my parents for all of their encouragement and support. TABLE OF CONTENTS ACKNOWLEDGEMENTS ............................ iii ABSTRACT ... .... .... ... .. .. .. .. ... .. .. ... vi CHAPTERS 1 INTRODUCTION ............ .................. 1 1.1 Approximation by Polynomials .......... ........ 1 1.2 Lagrange and HermiteFejer Interpolation ............. 5 1.3 Birkhoff and BirkhoffFej6r Interpolation ............. 10 1.4 Markovtype Inequalities .......... ............ 12 2 BIRKHOFF INTERPOLATION : (0,1,3,4) CASE ........... 16 2.1 Preliminaries.... ... .16 2.2 Existence and Uniqueness ........... .......... 18 2.3 Explicit Representation ........... ........... 21 3 CONVERGENCE RESULTS FOR A BIRKHOFFFEJER OPERATOR 28 3.1 Preliminaries and Convergence Theorem. . ..... 28 3.2 Estimate of the Fundamental Polynomials of the Fourth Kind .30 3.3 Estimate of the Fundamental Polynomials of the Third Kind .39 3.4 Estimate of the Fundamental Polynomials Ci(x) and C,(x) .41 3.5 Estimate of the Fundamental Polynomials of the Second Kind 43 3.6 Estimate of the Fundamental Polynomials of the First Kind 45 3.7 Proof of the Convergence Theorem . ..... 47 4 ERDOSTYPE INEQUALITIES ..................... 49 4.1 M ain Results . ... . 49 4.2 Some Lemmas ...................... ...... .. 50 4.3 Proofs of Theorems ........................ .... .. 55 5 TURANTYPE INEQUALITIES ..................... 62 5.1 Main Results .......................... 62 5.2 Some Identities ............................ 64 5.3 Proofs of Theorems ....................... 70 6 SUMMARY AND CONCLUSIONS ..................... 78 6.1 Synopsis . . . ... 78 6.2 Open Problems ............. ...... ......... 79 REFERENCES ..................... .............. 81 BIOGRAPHICAL SKETCH .................... ........ 85 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SOME PROBLEMS IN APPROXIMATION THEORY By Brandon Underhill May 1996 Chairman: Dr. Joseph Glover Major Department: Mathematics We begin by providing an historical background and some results concerning polynomial approximation and interpolation. Next we consider Birkhoff, or lacunary, interpolation and its development. Then we provide the historical basis and develop ment of Markovtype inequalities, and related best constant problems when the class of polynomials is restricted in some way. We first investigate the (0,1,3,4) case of Birkhoff interpolation where the nodes of interpolation are the zeros of the integral of the Legendre polynomial. We prove the existence and uniqueness in the 'modified' (0, 1,3, 4) case, and then provide an explicit representation in this case. Next we prove that the 'modified' (0, 1,3,4) BirkhoffFejer operator (based on the zeros of the integral of the Legendre polynomial) converges uniformly for the entire class of continuous functions on [1, 1]. This provides only the second known case of such a BirkhoffFejer operator the first being the (0,3) case, both 'modified' and 'pure', studied first by Akhlaghi, Chak and A. Sharma who proved the existence and uniqueness and provided explicit forms for the fundamental polynomials, and then by J. Szabados and A.K. Varma who provided a new representation for the fundamental polynomials of the first kind, and proved the convergence results. Let L, denote the Lorentz class of nonnegative polynomials of degree n on [1, 1]. In 1940, P. Erdos proved a refinement of Markov's inequality for polynomials with all real zeros which are outside (1, 1). We extend the results of P. Erdos, P. Erdos and A. K. Varma, and G. V. Milovanovi6 and M. S. Petkovi6 for P, E Ln in the L2 norm with the ultraspherical weight w(x) = (1 x2)a, a > 1, and we extend these results in a weighted L4 norm. Let now H, be the set of all polynomials of degree n whose zeros are all real and lie inside [1, 1]. We provide the lower bound analogues to the Erd6stype inequalities for P, E H,, as well as extend the results of P. Turin and A. K. Varma with an asymptotically sharp result in the LP norm for p an even integer. We conclude with a summary of the results and note some related open prob lems. CHAPTER 1 INTRODUCTION 1.1 Approximation by Polynomials In 1715, the English mathematician Brook Taylor (16851731) published his generalization of the Mean Value Theorem [46]. His method approximates a given ntimes differentiable function f by a polynomial P, of degree n in (x a) required to satisfy the conditions (1.1.1) P.k)(a) = f(k)(a), k = 0,..., n. These conditions yield the polynomial P(x) = k(x a)k k=0 the socalled nth Taylor polynomial. Taylor used these polynomials to approximate solutions to equations. Let us denote by C[a, b] the class of continuous functions on the interval [a, b]. Later on we shall have use for the big '0' notation, whereby O(k) means less than or equal to a positive constant times k. Let us denote by HI, the class of all algebraic polynomials of degree at most n, and denote the uniform norm for f C[1, 1] IlfI = max If(x)l. 1 Lagrange proved the following version of Taylor's Theorem with remainder. Theorem 1.1 If f and its first n + 1 derivatives are continuous on an open interval (c, d) and if x and a are points of (c, d), then f (x) = P,(x) + f(( a) (n + 1)! where 0,+1 is some number between x and a. Unfortunately, the Taylor polynomials require a function f to be ntimes differentiable, and still may yield poor approximations to f outside a very small neighborhood of a, as some derivatives of f at a may be very large compared to f(a). Also, they are not a very efficient way to approximate a function. For example, the error in the approximation for f(x) = ex by the third Taylor polynomial P3 about a = 0 on the interval [1, 1] can be seen to be lie P3(x)I I 0.0516, where the error is not evenly distributed through the interval. As is typical of ap proximations by Taylor polynomials, the error is much smaller near the origin than near the endpoints 1. In 1885, K. Weierstrass [55] discovered a theorem that founded a theory of the approximation of functions, which can be stated as follows. Theorem 1.2 If f is in C[1, 1], then there exists a sequence of polynomials P, such that P, > f uniformly on [1, 1]. The socalled Bernstein polynomials B(f,x) = (2knn)( (1 + x)k(l x)nk k k!(n k)!' k=0 provide such a sequence. Observe that if f 0, then Bn(f,x) > 0. Thus, we say that Bn is a positive operator. It turns out that if f 2 0, then B, is an element of the Lorentz class of polynomials, which we shall investigate further in Chapter 4. The Bernstein polynomials also have the nice property that for any given rtimes continuously differentiable function f, we have B'j)(f, x) + f()(x) (j = 0,...,r) uni formly on [1, 1]. On the other hand, the convergence of the Bernstein polynomials is generally very slow. For example, if we choose f(x) = 22, then lim n[Bn(f,x) f(x)] = x(1 z), n+oo so that for large values of n we have BIIB(f,x) x211 =0(1x(1 x)) (n Thus, the error does not decrease rapidly with n, even though f is a very simple function. Let us digress for a moment to consider how Bernstein may have come up with these polynomials. Bernstein knew probability theory, and probably reasoned as follows. First, suppose that the probability of an event occurring is x, where 0 < x < 1, and so the probability of the event not occurring is 1 Now, the probability of the event occurring precisely k times in n attempts is given by )xk(l k) It follows then that n n (n )xk(l )n = (sum of probabilities) = 1. k=O k=O One may also observe 1= [(1 x) + ] = k ) l ) nk k=O Suppose now f E C[0, 1] and x is chosen randomly in [0,1]. For a given positive integer n, consider the set {f( ) : k = 0,... ,n}. If n is large enough, then at least one of the numbers f( ) lies close to f(x). We want to find a weighted sum Ew(k, x)f (), w(k,)= 1, k=O k=O that yields a good approximation to f(x). It follows from the Law of Large Numbers in probability theory that choosing the weights w(k,x) = ( )Xk(l x)nk, the polynomials f ) k( X)nk k=0 converge uniformly to the function f on the interval [0, 1]. These are simply the Bernstein polynomials on [0, 1]. Once Bernstein found these polynomials, he gave a proof of uniform convergence without using the Law of Large Numbers. As a tool to measure the rate of convergence, we now introduce the classical modulus of smoothness (or continuity) of order s w,( f,) = sup IA'fl, 0 where ALf(x) = f(x + h) f(x h), Af (x) = Ar f(x + h) Arhf(x h). It has been shown by G.G. Lorentz [21] that the error in the case of the Bernstein polynomials is IB.(f) fj = OW /,f , and this is the best possible. As C[1, 1] is a normed linear space, for a given f E C[1, 1], there always exists a polynomial of best approximation to f. In fact, for each n there exists a unique polynomial p IIE such that IIf P~Il < IIf pll for all p IIE . Set E,(f) = IIfp;II. It has been shown by D. Jackson [19] that E,(f) < 6wi (f, ). Because it is in general very difficult to obtain the polynomial of best approximation p,, one often considers the best least squares approximation. That is, one minimizes lf p, IL2 where IfL2 = (L w(x){(f(x))2dx) 1 and where w(x) 2 0 and f w(x)dx exists. These approximations are usually pretty easy to compute, typically yielding a good uniform approximation to f, superior to that given by the Taylor polynomials. For example, we have for the function f(x) = ex that the least squares approximation r*(x) (w(x) 1) of degree 3 gives the error Hex r*(x)ll 0.0112. The error for the polynomial of best approximation p*(x) of degree 3 here is lie p;(x)lI 0.0055. The weight function w(x) allows for different levels of importance to be given to the error at different points in the interval. Later, in Chapters 4 and 5, we shall investigate some inequalities for polynomials in such a weighted L2 norm. 1.2 Lagrange and HermiteFeijr Interpolation Let us now consider the following problem. Given n distinct nodes 1 < < n1 < < and data yi,y2,... y,, find an algebraic polynomial of least degree whose graph passes through the points (xk, yk) for k = 1,...,n. That is, find a polynomial, say Ln1, such that (1.2.1) Lni(xk) = yk, k=l ,...,n. We denote the Kronecker delta function Sk, by Sk 1, if v = k, kv{ 0, if v k. Now, if we could find polynomials (x) (v = 1,.. n) such that (1.2.2) t,,(xk) =k, k = 1,...,n, then we could write (1.2.3) n Ln1(x) = yv(W), v=i and the polynomial Ln1 satisfies (1.2.1). But, ea(x) = is a polynomial satisfying (1.2.2). Thus, we have shown the existence of a polynomial Ln1 E n1 satisfying the conditions (1.2.1). Suppose now that there exists another polynomial pn1 E IIn that also satisfies (1.2.1). Then Lnl(Xk) Pn(Xk) = 0, k = 1,...,n, and Ln (x) Pn (x) is a polynomial of degree at most n 1 having n zeros. Then L,_(x) pn(x) = 0, or equivalently, Ln1 = pn1. Thus, there always exists a unique polynomial of degree at most n 1 satisfying (1.2.1), and it is given by (1.2.3). We have just derived the socalled Lagrange interpolation formula. The polynomials ,(x) are called the fundamental polynomials of Lagrange interpolation. Lagrange was interested in using interpolation to exploit the information in astronomical tables, and around 1790 he presented a paper to the Academy of Sciences in Berlin. If we define w(x) = (x Xi)(x x2) . (x x,), then the numerator of e,(x) can be written as (2) Observe then that W'(xh) = (xv X1) .. (X xo1)(x X"+i) ... (X, Xn) is the denominator of ,(x). That is, we can write ,(X) = '(x ) (X X,)W'(xV) This is a more useful form of the fundamental polynomials of Lagrange interpolation. Concerning the error in Lagrange interpolation, Cauchy showed that for a function f that is ntimes differentiable, we have f(x) L, (x) = f n ( W for some number ( E (1,1). Notice that if f II,,1, then f(n) = 0, and we have Ln1 f. We say then that Ln1 is a projection operator. One might think it reasonable to expect that if a system of equally spaced nodes are chosen, then for a continuous function f on the interval [1, 1] we should have Ln + f uniformly on [1, 1]. In 1901, C. Runge [30] presented his classical example that this is not necessarily the case. Let Xk = 1 + (k for k = 1,...n, and choose f(x) = Runge showed that the Lagrange interpolation polynomial does not converge to the continuous function f on [1, 1]. In fact, he showed that for 0.72 < xj < 1, we have lim Lni(f,x) = 0o. noo This result is rather disappointing, and in fact in 1914, Faber [13] published his result that for every choice of nodes nE \(X) > log n \ ( \ >  Unfortunately, this means that for any system of nodes, one can always find a function f such that Lnl(f) becomes unbounded. That is, there is no universally effective system of nodes. A proof of this result can be found in the book of Rivlin [29]. Let us define now the Lebesgue constant n An1 = max if(x)I. 1 It is easy to see that f Lllj Eni(/f)(1 + An1) < 6(1 + A_,1)W (/, ) \ n' Thus, for given f E C[1,1] and a given system of nodes, L,_(f,x) + f(x) uni formly if AnIw (f, ) + 0. In looking for interpolation polynomials which are uniformly convergent for the whole class C[1, 1], L. Fej6r [14] considered the socalled Hermite [17] interpolation polynomial H2ni E 112n1. Here, given data yl,...,y, and y',...,y', we require that H2n1 satisfies the following conditions H2nl(Zk) = Yk, Hn(xk) = y', k =1,...,n. Thus, on any set of real distinct nodes, the Hermite interpolation polynomial has the form n n tt2n x) = ZykAk(x) + >Ey[Bk(x) k=l k=l where the fundamental polynomials are given by Ak(x) = e (x)[1 2 '(xk)(x xk)], Bk(x) = (x Xk)(), k = 1,...,n. In the case that the nodes are the zeros of the Chebyshev polynomial T,(x), we have that Ak(x) = (x) ( >0, and it follows that n SIAk(x)l= 1. k=l Fej6r showed that for any continuous function f on [1, 1], the operator n R2nl(f,x) = f(xk)Ak(x) k=1 converges uniformly to f on [1, 1]. Two proofs of this result, as well as a nice discussion, are given in the paper of T.M. Mills [27]. Note that this operator has the properties that R2ni(f, k) = f(xk) and R'n1(f, k) = 0, k = 1,...,n. Thus, when all the higher derivative information is set equal to 0, we refer to such a polynomial as a HermiteFejdr operator. This was one of the first interpolatory proofs of the Weierstrass Approximation Theorem. It has been shown that for f E C[i, 1] we have IiH2.1(f) f\ = o (f), l and this is the best possible in the sense that for the function g(x) = Jxz we have HIni(g2 ) l> c ( ,10 n). One can generalize the Lagrange and Hermite interpolation problem to the socalled general Hermite interpolation problem (or simply Hermite interpolation). In this problem we seek a polynomial Hnmi satisfying the mn conditions Hmi(xk) = y j = 0,...,m 1, k = 1...,n, where the numbers yk are given data. Notice that when n = 1, Hnm is just the (m 1)t Taylor polynomial. The (0,1,..., m 1) Hermite interpolation problem always has a unique solution Hmi E n,,m1 on any set of real distinct nodes. Despite the positive result in the (0, 1) case of Hermite interpolation, J. Sz abados and A.K. Varma [42] showed that the Lebesgue constant in the (0, 1,2) case for every choice of nodes has the property n A3,1 = max IAo,3,k (X)l > clogn, S k=l where the Ao,3,k(x) are the fundamental polynomials of the first kind of (0,1,2) inter polation. Thus, one cannot obtain uniform convergence for the whole class C[1, 1] for the (0, 1,2) HermiteFejer operator for any choice of nodes. Later, J. Szabados [40] showed that n m1 Hn_(f,x)= E y)A ), k=l j=O where the fundamental polynomials of the (j + 1)t3 kind Ajk(x) of (0, 1,..., m 1) Hermite interpolation are given by Ak ) = k Jk(X Xk)i+j, j = 0,...,m 1 k=l,...,n, Ajk(X) = j i! ' i=O and he showed that for every choice of nodes Aj)lkl> .1 (cn) if m j is odd, Sif j is even. k=l nJ In particular, when m is odd, we have for any system of nodes n II Ao,k(x) >clog n k=l so that there cannot be uniform convergence for the whole class C[1, 1] for the (0,1,..., m 1) HermiteFej6r operator (m odd) for any choice of nodes. 1.3 Birkhoff and BirkhoffFeijr Interpolation In 1906, G. D. Birkhoff [8] considered the interpolation problem where in formation is prescribed for higher derivatives which are not consecutive. In this case, unlike Hermite interpolation, a unique solution does not always exist. The (0, mi,..., m,_l) Birkhoff (or lacunary) interpolation problem consists of finding a polynomial Qs, such that QJ(xk) = y k= 1,...,n, j= 0,m,...,ms1, where the yj) are given data and 0,mi,...,m,_ are not all consecutive integers. We see that Q, must satisfy sn conditions, so that Qn E IIan_1. In general, it is very difficult to find an explicit representation of these polynomials. In fact, it is usually difficult to even determine when there exists a unique solution to this problem. Let Pn(x) denote the nth Legendre polynomial normalized by Pn(1) = 1, and define irx,() = (1 x2)PI(x). In 1955, J. Surinyi and P. Turn [39] began studying the case of (0,2) interpolation on the zeros of ir,(x). They showed that for n even, the (0,2) interpolation problem has a unique solution. Later J. Balizs and P. Turn [5] provided explicit forms for the fundamental polynomials. They also proved convergence results and Markovtype inequalities with these polynomials. The condition of convergence in this case was later improved by G. Freud [15] and H. Gonska [16], but P. VWrtesi [54] has shown that the process is not uniformly convergent for all continuous functions, as the Lebesgue constant of this type of interpolation is of order exactly O(n). In 1958, R.B. Saxena and A. Sharma [34],[35] extended the results of Turin to (0, 1,3) interpolation, and later Saxena [32] extended them to the (0, 1,2,4) case. In 1962, Saxena [33] handled the case of 'modified' (0, 2) interpolation on the same nodes. Here, by 'modified', we mean that instead of prescribing second derivative information at the endpoints +1, we prescribe first derivative information there. We note that in general, 'modified' cases are more easily handled, and lead to the solutions in the 'pure' cases. A. K. Varma, R.B. Saxena and A. Saxena [52] studied the case of 'modified' (0, 1,4) interpolation (second derivatives instead of fourth derivatives are prescribed at 1) on the above zeros, showing the Lebesgue constant to be O(log n). Thus, they conjectured that this process cannot converge for the whole class of contin uous functions on [1, 1]. Recently, A. Sharma et al. [38] have shown that the Lebesgue constant for the modified (0, 2, 3) interpolation (first derivatives instead of third derivatives are prescribed at 1) on these zeros is also O(log n). In looking for Birkhoff interpolation procedures which converge uniformly for all continuous functions on [1, 1], J. Szabados and A. K. Varma [43] considered higher order (0, M) interpolation. Specifically, they were able to show that the (0, 3) BirkhoffFejer operator on the above zeros converges uniformly for all continuous functions on [1,1]. More precisely, they proved that for f E C[1, 1], the polyno mial R,(x) E IzI2n satisfying R,(xk) = f(k), R`(xk) = k= 1,...,n has the property that If(x) R(x)ll =0 ( (f,o ) where the xk are the zeros of ir,(x) and w3(f, 6) is the third modulus of smoothness of f. Akhlaghi, Chak and Sharma [2] had already proved the existence and uniqueness and provided explicit forms for the fundamental polynomials for the (0, 3) case, as well as the (0,2,3) case [1]. We note that the 'modified' (0,2), (0,3) and (0,2,3) cases on the zeros of 7r, (x) have been generalized [38] to the 'modified' (0,...,r 2, r), (0,..., r 3, r) and (0,..., r 3, r 1, r) cases, respectively. Given the positive result in the (0,3) case (and the negative results of the others previously mentioned) we turned to consider the situation where we have a similarly 'balanced' process. In particular, using an alternative representation to that given by Sharma et al. [38], we show that the Lebesgue constant for the 'modified' (0, 1, 3,4) interpolation (second derivatives instead of fourth derivatives are prescribed at 1) on the above zeros is 0(1). This enables us to prove the uniform convergence of the 'modified' (0, 1,3, 4) BirkhoffFej6r operator for the whole class of continuous functions on [1, 1]. 1.4 Markovtype Inequalities In 1889, A.A. Markov [24] proved that for any polynomial P, of degree < n on the interval [1, 1] we have (1.4.1) 11^p11 < n21llp11, where equality holds only for P,(x) = cT,(x), where T,(x) is the nh Chebyshev polynomial. The Russian chemist Mendeleev [26] had settled the question for the case n = 2 in his studies of the specific gravity of a substance as a function of the percentage of the dissolved substance. In 1892, A.A. Markov's brother W.A. Markov extended the Markov inequality to all higher derivatives (published in German [25] in 1916), providing an inequality sharp for every n. In 1912, S.N. Bernstein [6] improved Markov's inequality by providing the following pointwise estimate (1.4.2) IP'(x)l < l P 11. Note that this provides a much sharper inequality, except near the endpoints 1. Both the Markov and Bernstein inequalities play a key role in proving convergence theorems, as we shall see later. The inequalities of A.A. Markov and S.N. Bernstein can be improved if the class of polynomials is restricted in some way. Let us denote by Sn the set of all polynomials whose degree is n and whose zeros are all real and lie outside (1, 1), and denote by L, the set of all polynomials of the form (1.4.3) P,(x)= Eakqnk(x), ak >0 (k = 0, 1,..., n) k=0 where qnk(X) = (1 + X)nk(1 X)k. In 1940, P. Erd6s [10] proved the following refinement of Markov's inequality. Theorem 1.4.1 (P. Erdos, 1940) Let P, E Sn. Then we have 1 11P'11 < 21enj1P 1, 2 where the constant 'e cannot be replaced by a smaller one. In 1937, E. Hille, G. Szego and J. D. Tamarkin [18] had extended the Markov inequality in the LP norm, showing there exists a constant A such that for every algebraic polynomial Pn(x) of degree n we have 1 P f p (x) P dx) < An2 ( IPn(x)Pdx) \Jl \Jl where p > 1 and A is independent of n and Pn(x). Further, they noted that the problem of obtaining the best constant in the above problem is extremely difficult. In 1986, P. Erd6s and A. K. Varma [11] settled the above inequality for the Lorentz class Ln of polynomials in the L2 norm as follows. Theorem 1.4.2 (P. Erd6s and A. K. Varma, 1986) Let P, E Ln, n > 2. Then we have (1.4.4) f'I < n(n 1)(2n + 1) (pf' (P(x))d 4(2n (3) n))2d with equality if and only if Pn(x) = c(1 x)n1(1 T x). Also, if P, E Ln, then we have (1.4.5) (1 x2) (P(x)) 2d < + 1)(2n + 3) (1 x2)(Pn(x))2dx 1 4(2n + 1) i with equality if and only if P,(x) = c(1 x)n. It is known [22] that if Pn E Sn, then Pn E L, or P, E Ln. Thus, (1.4.4) can be viewed as an extension of Theorem 1.4.1 in the L2 norm. In 1988, G. V. Milovanovi6 and M.S. Petkovi6 [28] extended (1.4.5) with the ultraspherical weight (1 x2)", a > 1 (a > 1 if P,(1) = 0). A new proof is provided, and the requirement that P,(1) = 0 is removed. Then an extension is provided in a weighted L4 norm. Let now Hn denote the set of all polynomials of degree n whose zeros are all real and lie inside [1, 1]. In 1939, P. Turn [45] proved this analogue of Markov's inequality. Theorem 1.4.3 (P. Turn, 1939) Let Pn H,. Then we have 11P.111 > 6 11P.III This result was later sharpened by J. Erod [12]. In 1983, A.K. Varma [51] extended the above in the L2 norm as follows Theorem 1.4.4 (A. K. Varma, 1983) Let P, E H, and n = 2m. Then we have f'(P(x))2dx + 4 + +(n 1) (P()) where equality holds if and only if P,(x) = c(1 x2)m. Moreover, if n = 2m 1, then for n > 3 we have (P2(x))2dx + + 4(n 2)) 1(, ))2dx where holds equality if and only if P,,(x) = (1 x)m(1 :F x)m'. Earlier, A.K. Varma had given asymptotically sharp versions [48],[49] of this result, as well as proved the following [50]. Theorem 1.4.5 (A.K. Varma, 1979) Let Pn E H,. Then we have (n = 2m) (1 x2)(P(x))2dx > + 1 _(1 ( S 4(nl 1)+ (1  x2)(P,(x))2dx, where equality holds if and only if P,(x) = c(1 X2)". We extend this result in the ultraspherical weight w(x) = (1 x2)a, a > 1, (a > 1 if P,,(1) = 0). By a result of S.P. Zhou [56], for Pn, E H, and 1 < p < oo there exists a constant B, independent of n and Pn, such that ( IPn(x dx}p >BO I P dx) P We provide an asymptotically sharp result for p an even integer, showing in the limit that B (p 1)(p 3). 53 p2 CHAPTER 2 BIRKHOFF INTERPOLATION : (0,1,3,4) CASE 2.1 Preliminaries The objective of this chapter is prove that the problem of 'modified' (0,1,3,4) interpolation on the zeros of the polynomial 7r,(x) has a unique solution, and to pro vide an explicit representation in this case. By 'modified', we mean that instead of prescribing fourth derivatives at the endpoints +1, we prescribe second derivatives there. First we take care of some preliminary items. Let P,(x) denote the nth Legendre polynomial normalized by Pn(1) = 1. Let (2.1.1) 7r,(x) = (1 x2)P'_,(x) and enumerate the zeros of 7rn(x) by (2.1.2) 1 = Xn < Xn_ < ... < Xi= 1. We list the following known identities. These can be found in the book of G. Sansone [31]. (2.1.3) [(1 x2)P_, (x)]' + n(n 1)P,_1(x) = 0 (2.1.4) xP_1(x) Pn_2(x) = (n 1)P1(x) (2.1.5) (1 x2)P _(X) = (n 1)[Pn2(x) xPn_(x)] (2.1.6) P'(x)) P_2(x) = (2n 1)Pnl(x) (2.1.7) (1 x2)7r(x) + n(n 1)rn(x) = 0 (2.1.8) (x) (x 7r,( ) x x)7r' (x,) (2.1.9) I'( }) (x x )i^k)(x)+ ketkl)x)^ and we make note of the following values (2.1.10) Sn(n 1) _l(1) = n(n Sn (n 1)2 2 r'(1) = n2(n 1)2(n + 1)(n 2) Sn(n 1) 4 (n + 1)n(n 1)(n 2) = 24 1 (n + 2)(n + l)n(n 1)(n 2)(n 3) 192 (2.1.13) )X n(n 1) S 5(1 x)2(n n + Observe from (2.1.7) that 7r"(x,,) = 0, n(n 1) 3(1 x) 24 18 1 ), v= 2,...,n1. v= 2,...,n 1. (2.1.11) p(n + 1)n(n l)(n 2) Pn ) = 8 (1 )=n(n 1) (2.1.12) (x,) = 0 '(n(n 1)x, S (1 x )2 2.2 Existence and Uniqueness We shall prove the following. Theorem 2.1 Let a,, b, d,,, v = 1,..., n; cl, cn, and e,, v = 2,..., n 1 be given real numbers. Then there exists a unique polynomial Qn(x) of degree < 4n 1 such that Q.(x,,) = a,,, Q'(x,) = by, V = 1,... ,n Qn(1) = c1, Q"n(1) = Cn, (2.2.1) Q'(zX,) = d,, = l,...,n Q()(x,) = e,, = 2,...,n 1. From linear algebra, this is equivalent to proving that if Qn(x) is a polynomial of degree < 4n 1 satisfying (2.2.2) Qn(X)= Q',(x) = Q'(x,) = 0, = 1,...,n Q1(+I1) = 0, Q(4)(x,) = O, I= 2,...,n 1, then Q, = 0. Before we proceed with the proof, we note that for the polynomial rv2(x) we have (2.2.3) 7r (x,) = [7r?(x)]' = 0, [(r(x)] = 27r(x,)2, = 1,.. n, [r 1in 2 (4) 8n(n l)7r^(a; )2 [lx()] = 0, [7L(x)]L = 12 =2,...,n1. Proof of Existence and Uniqueness Suppose that Qn H4,Il satisfies the conditions in (2.2.2). We show that then Qn = 0. Since Qn(x,) = Q'(x,) = 0 for v = 1,...,n, we write Q(x) = 7r2(x)q2,_1i(), where q2n1 E I12n1. Then we have Q(1) = r(1)q2n_(1) + 2[ r(x)] 1q_.(1) + [7r (x)]1q2n1() = [(x)]lq2nl(1). As Q"(1) = 0 and [7r'(x)]+ = 27r(11)2 : 0, we have q2n1(l) = 0. 7r2 2n xl[7r2(X), [ 3rr2(Ir x (x) I" q2n I (X.) Q"1(X,) = 7 (x,)q"2Nn(,)+3[1 ()]'q1'+ 3 ),Q 2n1 v 92n)] 1 & = 3[,( (x)'mq2,_ (x,), v = 1,...,n, 3nd 2sincel~q, _J(X.,) and since [r2(x)]" = 27r'(x,)2 7 0, we have q2n1() = 0 for = 1,...,n. From the paper of Aklaghi, Chak and Sharma [1], page 63, we have q2n1 IIn2, and there exists sni E IIl such that (2.2.4) q2nl(x) = 7rn(x)sn1(x) 7'(x)SnI(x) where s,_i(l) = 0. Using (2.1.7), we have (2.2.5) Sn(n 1 14V Thus, qn'(x,) = r(xX)s"_1(x.) 7r"(x,)S,l(x,) 7r(x,,) ((1 x)s(,) + n(n 1)n()) . Now, () = 6[r(x)]q~~ n_(x,,) + [r(x)]) q2n l(x,) 12= () ((1 x)sI_(x) + n(n 1)snX,)) 87r' (,)3 + n(n 1)s, (x) = 47 (3(1 x )s" (x) + 5n(n l)s1(x,)) = 0, v =2,...,n 1. As 7r(x,,) 4 0, we have (2.2.6) 3(1 X)sn_l(x,) + 5n(n 1)sn_l(xz) = 0, v = 2,..., n 1. In fact, since sn,(l) = 0, (2.2.6) holds for v = 1,..., n. Then 3(1 x2)s"_,(x) + 5n(n 1)snl(z) is a polynomial of degree at most n 1 having n zeros. We conclude (2.2.7) 3(1 x2)s"_1(s) + 5n(n 1)sn(x) 0. Since s,_l(l) = 0, we can write (2.2.8) nl Sni(x) = Z z,7*r=(x). v=2 Replacing (2.2.8) in (2.2.7) yields n1 1[5n(n 1) 3v(v 1)]zr,(x) = 0 v=2 so that z, 0 for v = 2,..., n 1. Thus, s,_ = 0 and we deduce q2n1 = 0. This gives Qn 0. Theorem 2.1 follows. 2.3 Explicit Representation The polynomial Q, in the Theorem 2.1 will evidently have the form n n n n1 Q.(x) = E a,A,(x) + b ,B,(x) + ciCi(x) + nCn,(x) + d,D,(x) + > eE,(x) v=1 v=1 v=l v=2 where the uniquely determined polynomials A,(x), B,(x), D(x) (v = 1,...,n), (v = 2,... ,n 1) of degree < 4n 1 are characterized by the conditions (2.3.1) A,,(k) = 8ky, A',(k) = A'(Zk) = A"(1) = A"(1) = 0, A4) (k) = 0 3 (k= ,...,n) (k = 2,...,n1) B,(zk) = 0, B'(Zk) = 6kv, B k) = 0 (k = 1,...,n) B"(1) = B'(1) = 0, B(4)(xk) = 0 (k= 2,...,n 1) Cl(xxk C(k) = C'(k) = 0 C'(1) = 0, C1 (k) = 0 (k= 2,...,n 1) Cn(xk) = C'(xk) = C'(xk) =0 (k = 1,..., n) (2.3.2) (2.3.3) C'(1) = 1, (2.3.4) Cl(x), Cn(x), E,(x) (k = 1,..., n) (k =2,...,n 1) (2.3.5) D,(xk) D(x D'k) = 0, D"(1) = D"(1) = 0, Dv1 v D4) (k) = 0 (k = 2,...,n 1) E,,(k) = E,(xk) = E,(Zk) = 0 (k = 1,...,n) E"(1) = E(1) = 0, E4)(xk) = Sk, (k = 2,...,n 1) Theorem 2.2 The fundamental polynomials of the 'modified' (0, 1,3,4) interpolation based on the zeros of the polynomial 7rn(x) can be explicitly represented in the fol lowing manner. (2.3.7) E, (x) r(x)(1 x) n 4n(n 1)P _((,,) k=2 (2k )P'1(x,) 7(x)k() n(x(x)) k(k 1)Ak ( where Ak = 5n(n 1) 3k(k 1), k = 2,..., n 1, (2.3.8) D,(x) ,)3 4 n1 X) E Pn,(Xk)=2(Xk)Ek(X) (v = 1,.. ., n) C (x)e~(x) (1 x)wr(x)el(x) C1(x) = 2 4n(n) 2n2(n 1)2 4n(n 1) n 12 Pn(xk) (xk)2 k=2 3n(n 1) 2 n(n 1) 2 ( x 1) Ekx) Cn(x) = Ci(x) (2.3.6) (2.3.9) C )(Xk) = 0 C"(1) = 0, C",(1) = 1, D" (k)= (k = ,..., n) (2.3.10) n,(_ ) ((x) n(n 1)(13n2 13n + 1) Bi() (x 2n(n 1)C(x) n(n )(13n2 13n + 1D(x) n(n 1) 8 n1 24 P_l (Xk).i (xk)3Ek(x) k=2 Bn(x) = BI(z) (2.3.11) BL ,(x) + 7rn (x.) 4n(n 1) D 13n(n )1z, 1  22 E(2) 1 v( i + 4 n1 Pn1(x(,) kE (v = 2,...,n 1) (2.3.12) Ai(x) = e(x) n(n 1)Bl(x) 4(3f'(1)2 + l(1))CI(x) n1 4(6e+(1)3 + 91 (1)D'(1) + "(1))Di(x) 24 )(xk)4Ek(x) k=2 An(x) = Ai(x) (2.3.13) 4n(n 1)xzvD A,(x) = (x) + )D,x(z) + (1 x )I n1 24 ,(xk)4Ek(x) k=2 24n(n 1) ( 5(1 x)2 n (v = 2,...,n 1).  n ) E,(x) 1 v Proof of Theorem 2.2 We first provide the proof of the representation of the last fundamental poly nomials. Following the proof of Theorem 2.1, we write E,,(x) = i7r(x)q2n i(x) where q2ni(x) = 7rn(x)s'_1(x 7r'(x)sn1(x) E H2n2. Then, instead of (2.2.6) we get 4r(x (3(1 X2 )i(k) + 5n(n l)n(k)) = k,, k = 2,..., n 1, 1 Xk or equivalently, 1z 1 a2 3(1 x )s"1(k) + 5n(n 1)Sn1(Xk) = 4(X,(k), k= 1,...,n. 47ra(X,,)3 Hence, 1 Z (). 3(1 x2)s"1() + 5n(n 1)s,_1(x) 4 ,(r(). Now, using the identity 1 2 1 ,(x) = n(n  1)Pn ) L k 1 Pk(X)( v = 2,...,n 1, which can be found in the (0,2,3) paper of Akhlaghi, Chak and Sharma [1], page 58, and using the representation (2.2.8) we obtain n(n 1)(i x2)(2k 1)P_,(x,) zk 4k(k 1)Akr(x)5 so that n(n 1)(1 x2) 1 (2k 1)P_1(xX) 47r'(x2,)5 = k(k 1)Ak and we have (2.3.7). To verify (2.3.8) we observe (2.3.5), (2.3.6) and that for the polynomial 7rn(x),,(x) we have S(xk)4(Xk) = [ (),(x) = [7r(x),(x)]k = 0 k = 1,..., n n \ kcXj ~\/ C\ k kLl\lVu 2 ['(),( )] = 6r (x)36k k = 1,...,n 47r(xk)3 f'2k) [l (x)4(x)] = 3( ) 7T(.A) k= 2,...,n 1. To verify (2.3.9), we observe (2.3.3), (2.3.5), (2.3.6) and that for the polyno mial 2n(n 1)2 2n2(n 1)2 (1 x)n(x)l(x) 4n(n 1) 2n2(n 1k) 2n2(n 1)2 + (1 xk)r (xk) 2(xk) 4+ 0 , 4n(n 1) I (x)x) (1 x) (x)' 2n2(n 1)2 4n(n 1) J k 1 r(x)e~(x) (1 xr())]"= 1, .2n2(n1)2 4n(n 1) J1 2f() ((x) (1 z)7r(2) (x) " S2n2(n 1)2 4n(n 1)r)" 2n2(n 1)2 4n(n 1) 1 2n2(n ()2 2n2(n 1)2 (1 x)7r(x)(x)]" 4n(n 1) Jk 2n2(n 1)2 n(1 n ) (x)() (4 4n(n 1) 1k 127r (xk)2e (xk)2(1 _ (1)) n2(n 1)2 k =2,..., n 1. we have 3n(n 1) 2 k = 1,..., n, To verify (2.3.10) we observe (2.3.2)(2.3.6) and that for the polynomial 7r,(tx)t3() we have 7r(xk)e(Xk) = 0, [7M(X) 1(Xs)]' = 7r(1)6k1 = n(n 1)Ski, [7rn(x)(x)()]k = 2n2(n 1)2k, n2(n 1)2(13n2 13n + 1)k 8 k= 1,...,n, [n(X)f(X)1 = 24 (X)el(k k) , k=2,...,n 1. To verify (2.3.11) we observe (2.3.2), (2.3.5), (2.3.6) and that for the polyno mial 7rn(x)e~(x) we have 7n(xk)t(xk) = [7n(x)e~(x)]'k = 0, k = 1,...,n 4n(n 1)7r(x(,) [7rnk(X).(X)K =J1 Sk [7rn(x)tyi(x)] )= 24r,(xk),(xk)3, k = [ (7r, )( 4) 13n(n 1)xzxr(x1,) (1 )2 k :v, rn(X)'(X)]'x = [n(X) (X)];" = 7 (X,)Sk, To verify (2.3.12), we observe (2.3.1)(2.3.6) and that for the polynomial f(x) we have I(zk) = Jkl, [e(X)l = 4Ve(1)Skl = n(n 1)Skl, k = 1,...,n, [(=(X)]" ( (12i(1)2 + 4l(1))sk1 n(n 1)(11n2 1n 4), 12 [f (x)].. = (24t'.(1)3 + 36f (1)'(1) + 4i"'(1))Sk1, [t(x)]~1 = 24e1(xk)4, Finally, to verify (2.3.13) we observe (2.3.1), (2.3.5), (2.3.6) and that (xk) = Sk, [e( [t(z)]i, = 0, [ ,(X)]" = [j(x)] ) = 24',(xk)4, )]k= 0, k = ,...n, 4n(n 1)x (1 x~~ k k (1 Sl)2 ,24n(n 1) 4 [e4(x)] ) = 36'(,)2 +4 )(,, ) = 24n(n2 ) 2 2) 5(1 2)2 1 X This completes the proof of Theorem 2.2. k = 1,...,n, k = 1,...,n, k= 2,...,n1. k :v, v = 2,...,n1. CHAPTER 3 CONVERGENCE RESULTS FOR A BIRKHOFFFEJER OPERATOR 3.1 Preliminaries and Convergence Theorem Let f be a realvalued function on the interval [1, 1], and define the 'modified' (0, 1,3, 4) operator R.(f,x) = Ef(,)A,(x), iv=1 where the fundamental polynomials of the first kind A,(x) (v = 1,..., n) are given by (2.3.12)(2.3.13). The main goal of this chapter is to prove the following. Theorem 3.1 Let f be a continuous function on the interval [1, 1]. Then IIf(x) Rn(f,x)I = (wl(, log n,)) where wl (f, 6) is the first modulus of smoothness of f. Observe that the rate of convergence is the same as that of the classical HermiteFej6r operator, but it is not as good as that of the (0,3) BirkhoffFej&r operator on the zeros of 7r,(x). We shall employ the following notations (3.1.1) x=cost, k = cost (k=l,...,n) (3.1.2) We shall use the following known estimates. (3.1.3) ( 1 2) (1 x2)7) Ir,(x)I = 0 (1 )i n) , These can be found in Szeg6's book [44]. (3.1.6) IPn(x) + Pn+1(x) = O sin t n 2 1 1< X < . 22 This is a result from the paper of Szabados and Varma [43], page 734. There exist absolute positive constants a1, a2, a3, a4 such that (3.1.7) 2 < 1 2 = 2 (n 1) x (n 1)2 2 2 (3.1.8) (n 1V)2 (n 1)2 (3.1.9) (n 1)2 p2 a3 (X) > nsint (3.1.10) Itv tV+l1 >  n (3.1.11) t+tv sin t < sin t + sin t, < 2 sin  , 2 t+t/ sin t < sin t + sin t, 2 sin  2 (3.1.4) (3.1.5) = [1]+1,...,n1, We shall refer several times to using an Abel transformation with factors bk, by which we mean that we are using the summation by parts formula q q1 E akbk = Ak(bk bk+1) + Aqbq + Aplbp, k=p k=p k where Ak = as. s=O 3.2 Estimate of the Fundamental Polynomials of the Fourth Kind Lemma 3.2.1 For the fundamental polynomials E,,(x) (v = 2,..., n 1) we have the following estimate n6 SiE = O sin2 2 ) if t,I> > 0 sint in3 t sin4 tv \ _t\ IEo(x)I 0sin i n sin2 n , O if It ti~ < , where c > 0 is an absolute constant. Proof We shall first prove the case when t t,1 > . Using (2.3.7), (2.1.1) and (2.1.3) we have S(ixr(x)(1 x ) ,2) 7 ( 12k 1 E1,(x) 4n4(n 1)4P, ,) (1 x2)rk(k 1)Ak k1X)pkX) +X,(x) P.(x)Pk=2 n1 k=2 S1 + S2. Following the same argument given in the paper of Szabados and Varma [43], pages 736738, with Ak = 5n(n 1) 3k(k 1) we obtain n1 (2k 1) P'l(,)P '()P_(x) Sk=k(k 1)AXk n, (x Xv) S sin 2 + sin t, +0 tt3 2 2 n sin t sin2 t,, sin2 i sin ] 2 Thus, 7r (x)(1 X')(1 X')r,(X)P,'_(x)P'_(x,,) S1 S1 4n4(n 1)4Psl(x,)Anl(x x,) 2r(x)(i x2)(i x2)r,(x) ( sin_ , 4n4(n 1)4 P5_3(2,) .O2 4n(n 1) 1(x) n sin2 t sin t1, sin2 sin The first term in S1 will later cancel with a term occurring in S2. Using (2.1.3), (3.1.5), (3.1.9) and (3.1.11) we obtain the estimate O ( sin3 t,, n6 sin 112 sin4 t, +n6 sin2 v for the remaining term in S1. We turn now to estimate S2. Using an Abel transformation with the factors yields (3.2.1) 7 (X)(1 X2) "1 2k 1 S 4 = E 2k Pk1(x,)Pk1 () 4n(n 1)P() k=2 k 7r (X)(1 x) 1 n2 n( ) ) I 1 (2k + 1)Pk(x)P'(x,) 4n4(n 1)4p5_i(x) A1 k=1 ^.^^;i^[ n2 k=l 6 2 (2s k=2 1 + 1)P8(x)P.,(x.))]. Next, we differentiate both sides of the ChristoffelDarboux formula (3.2.2) (2r + 1)Pr(x)P(y) = (n + 1)P()P+() P+()P() r=O y X with respect to y and set z, equal to y to obtain (3.2.3) SP.(X) )P"'z(2) Pn+i()P'(x,,) E(2r + 1)Pr(x)PF(X,) = (n + 1) P+(x)P(x ) r=O P,(x)P.+1(x,) Pn+l(X)Pn(x,)] (x5 J)2 and applying this result to (3.2.1) yields n 1 Pn2(x)P(x,,) Pnl(x)P _2(x,) S2 =  _n1[ X P._2(x)P,(X,,) Pn.(x)P.2(X,) (X, x)2 n6 2 2 k Pkl(X)P (x)Pk(x)Pl(x) k k) _ Pk(x)Pkl(x1) 6 AAk+l\ x x (X, a)2 k=2 (n 1)P_(x)P'_2(x, ) (n 1)Pn2(x)Pni(x,) (n )P+(s)P,_2(,) An (x XL) Anl(x X,)2 An1(x x)2 6 n2 2 (Pl())P2)) 6( +Pk (X) Pk( x,) Pk(X)Pk(x,)) Sx zx, AkAk+l k=2 From (2.1.4) and (2.1.5) we have (3.2.4) P_2(x) = (n 1)P.n(x,) (3.2.5) Pn2(X ) = XPnl(z) (3.2.6) P,_2(x) 7n + zPnl(). n1 Using (3.2.4)(3.2.6) we combine the first 3 terms in S2 to obtain (n 1)Pn_(x)Px2(x,) (n 1)P,_2(x)P.i,(,,) (n 1)Pf (x)P,_2(x,) (nl(x xy) An,(Z X2)2 (nx(Z X,)2 (n 1)2P,_ (x)Pl) (n 1)P1(x,) (7r,(x) An1(x x,) An(x x,,)2 n 1 (n 1)xPnl(x)Pni(x,) Anl(x X,)2 (n 1)P,l(x)Pni(x,,) Pni((x,)rn,(x) Anl(x x,) An1(x ,)2 ' and multiplying through by 4 )4p5(. ) yields n(n 1)P,~ (x)Pn,(x,)r (x)(1 x) r(x)(1 x\) 4AXn(x x,,)n4(n 1)4p,_,(x2,) 4AXn1(x x,)2n4(n 1)4 P4_ (x)' where the first term above cancels with the aforementioned term in S1, and we estimate the second term above using (3.1.9), (3.1.5) and (3.1.11) to obtain ( sin4 t To complete the first part of the proof, we need only estimate the terms (3.2.7) 37r (x)(1 x) n"2 k2( x k(x)P ) 2(x x,)n4(n 1)4Pn,_ (xs) AkA k+l and (3.2.8) T2 = 3r (x)(1 ) Pk )Pk ) Pk )Pk 2(a X,)2n (n 1)4P5_(_ ) k= k As Pk(x) = ()kPk(x) and xn,+l = X,, v = 1,...,n, we may assume that 1 We show first that IT21 = 0 (n6 .2 We break this part of the proof into two cases. We begin with the case 1 < x < 1. Notice (3.2.9) Pki(x)Pk(2.)Pk(x)Pk1 (X) = I Pk(x)[Pk(X1 )+Pk(x)]Pk(X)[Pkl(X.)+Pck(X)] < IPk(x,)IlPk1l() + Pk(x)l + IPk(x)IlPk1(x,) + Pk (x)l, so that applying (3.1.3) and (3.1.6) yields (3.2.10) Pkl(x)Pk (x) Pk(x)Pk1(x,) < Ssin t 1 + Sn t, \k sln2 t. Thus, n2 k2 k 1 (Pk1(X)PkI(x) Pk(x)Pkl()) k=2 kc +l I t" n2 sin2 L sin2 L_ k =0 +  t k n4 sini t, n4 sin2 / k=2 (n2 sin2 t, sin2 t , I1 n2 sin t) +   . sin t, k sinz t Since ( 2n4(n 31(x)(1 x) 2n(n 1)4P (xx)(x S3 9 sin2 t sin2 t, n4 sin2  sin2 t+ 2 2) we get by using (3.1.11) sin2 t sin4 t, sin t sin5 t IT2 = 0 n6 sin2 1 sin2 tt + n6n sin2 t sin2 2 2 2 2 o sin4 tt, =0 n6 sin2 tt, 2 Now we consider the case when 1 < x, < 1. Since 1 . Applying this, (3.2.9), (3.1.5), (3.1.3), (3.1.9) and (3.1.11) to (3.2.8) yields the estimate IT1=o(n6 sin 4 t 50 nsin6 sin4 " Sn6sin2 t 2 ) in this case. Lastly, we show ( sin4 t, IT,1= 0 n6sin2 t_ 2 Using (2.1.6), we observe that (3.2.11) n2 k=2 n2 k2 = E AkAk+l (Pkl(x)P2(,) + (2k 1)Pk(x)Pk,(x,) Pk(x)Pk_(x)) k=2 n2 k2pkl(X)V) n2 k2 n2 = k k 2(X ) +k+ (2k1)Pkl(x)Pkl(X) k=2 kk+1 k=2 +k=2 k2Pk+ AkAk+i Now, combining the first and third sums above, and applying an Abel transformation k2 kAk (Pk1(2)P(X2) Pk(X)P_1(x5)) AkAk+n with the factors (k+1)2 to the second sum (after reindexing), we have (3.2.12) (n 2)2Pn2(x)P3(x) n Pk (x)P2(x) k (k )2 L An2n1 A k Ak+l Ak n3 + (An 2 k + 1)Pk()P k(x,) Anln2 k=1 n4 1)2 )2 ki + ( (k + 1)2 (k + 2)2 ) (2s + 1)P.(x)P.(x,)] k=2 ik+i k+2 k+2 k+3 s =1 We then apply the ChristoffelDarboux formula (3.2.2) and that k2 (k 1)2 (2k 1)(5n(n 1) + 3k(k 1)) Ak+l Akl Ak+lAk1 (2k 1)Ak Ak+lAkl to obtain (3.2.13) (n 2)2P,_2(x)P_3(x,,) (2k 1)Ak ,P An2n1 k=3 AkAlkAk+li S(n[ 2)3 PPn3()Pn2(X,) Pn2(X)Pn3(X,) An2An1 x (2k + 3)Ak+2 Pl(Pk )Pk(x,) Pk(x)Pkl(xv)) k1=2 k+lAk+2k+3 Xv  U1 + U2. Using (3.1.3) and (3.1.4), estimating termbyterm we obtain (3.2.14) Ul = On2 sin . n2 sin2 t sin2 t,, 37 We break the estimate of U2 into 2 cases. First suppose 1 < x, I< 1 Applying (3.2.10) to U2, and estimating termbyterm we get (3.2.15) IU = O sini t n sin2 sin 1 I n2Ix aj[sins2 t, Now, supposed < x, < 1 (from before we need only consider 1 < x < 0). Then 1 < 2, and we have on using (3.2.9) and (3.1.3) (3.2.16) (3.2.16) U21 = n2 t si t n2 sins2 tsm22 t) +0 (n4 E(2k + 3)k(Pk(x)Pk(x=) Pk(x)Pk2(x)) k=2 ( 1 Enk=(2k + 3) =0 .1 1 n2 sins t sin t, n sin2 t sin. ty, =0 2 1 1 n2 sin2 t sin2 t Thus, from (3.2.7), (3.2.11)(3.2.16), and then (3.1.9), (3.1.5) and (3.1.11) we obtain 37r(x)(1 x~) T1 = 2(x x,)n4(n 1)4P_ (X,) 1( sin? ti 1 i tsm2 t, 0 n2 1 S 3 2  (nsh fsin ^ n2si xt\ ssin' smn2 t Sn2 1 sin2 ti =0( sin4 tv = n6 sin2 tt This completes the proof of the case when It tI > . We obtain the second estimate in the lemma using the estimates I7r'(x)7rk(x) 7r,(x)7r' (x)I = 0 (nk~ , and (3.2.17) v=2,...,n 1. Thus, using the above and (3.1.5) and (3.1.9), we have when It t,I <  sin t sin2 tI =0 n2 n ()(1 XP ) n4(n 1)4'_P (x,) and 1(2k kl)Pk (x,) (7r'(x)rk(x) 7rn(x)7r(x)) Hence the proof of the lemma is complete. Hence the proof of the lemma is complete. Lemma 3.2.2 We have n1 E,(x)l O(n ). i (1 (1 2 1, 2  Sd^^ '" ) Proof By using Lemma 3.2.1 and (3.1.11) (n n1 S=2 sin t, sin 1^ __ 2 + n1 + n6 E sin2 t1t v=2 2 The two sums above can be found in the paper of Szabados and Varma [43], page = sin t =0 " n1 = O(n4).[ sin t = O(sin t,), 3.3 Estimate of the Fundamental Polynomials of the Third Kind Lemma 3.3.1 We have (3.3.1) IDI(x)l = O(n6), n1 ID .(x)l 1 2 v=2  X I= X2 v=2 1 ID.(x) = O(n6), ( log n =o , n3 , } = O(n ). We have on using (2.1.1), (2.1.8), (2.1.11), (3.1.4) and (3.1.5) 6n3(n (X) 6n 3(n  1)31 and using (3.1.3) and Lemma 3.2.2 we get 4n1 P1(Xk) E( 4 f( Ek(X k=2 (Xk 1) / n1 =0 n2 k=2 ) = (n 1 () Ek ) ck=2 /k IEk(x) (n), (1 )2 kI ) = r ) so that IDj(x)I = O(n6). Observing that D,(x) = Di(x) gives (3.3.1). We note next that for It t, I< on using (3.1.5), (3.1.9), (3.2.17) and that le,(x) < 1, we have (3.3.4) r(6) (n) 3((x) 6(1 x2)n3(n 1)3P3_(xL,) X= 0) (( 42 = 0(n S) 6(1 )(1 )n( 1)n 6(1 x)n3(n 1)3 (n ) (3.3.2) (3.3.3) Proof = (1 + x)P'I_(x)7r ( n) = O ( o( ) I 6n (n 1)< and further using (3.1.7)(3.1.8) we obtain (3.3.5) (x) e,(x) / (1 2)2nt)(l )n 6(1 x)2n3(n 1)3P_(x,) \ 6(1 x z)2n3(n 1)3 = O = O(n ). S6(1 )(n 1) O(n2) Now, applying (2.1.8), (2.1.3), (3.1.5), (3.1.9) and (3.1.11) we have (3.3.6) C 7r1w (X), S 6(1 x )n3(n 1)3P,_1 (x) Ittvl> 6(1 X2)n4(n 1)4P_i(x,)(x x,) 0 ( sint Ittvl> C v\' ~Vs" = (n 1)4 sinn It sin t Itt 2 2 (n sint (log n IttI>E sin n3 Next, using (2.1.8), (2.1.9), (2.1.3) and then (3.1.3), (3.1.9), (3.1.11) and Lemma 3.2.2 we obtain (3.3.7) n~ n( Sni) 4 E_ P21_P, _2) ZPl(Xk)et(Xk)Ek(x) J /=2 k=2 n_1 Pl_ (Xk) kX v=2 Itktk > SIEk(x) (1 x2)2 t Itktl>n (1  X)2p4_ (Xk) (1  x)(xk )P1 (x,) I Ek(x)j k(1_ x2)2 L SItk t > = O nlogn1 E(1 X 2)2 = log n) =O( n3 0('). We note that when v = k, the double sum in (3.3.7) is 0 as ef(x,) = 0, v = 2,..., n1. Observing (3.3.4), (3.3.6) and (3.3.7), we see that (3.3.2) holds. To show (3.3.3), we observe (3.3.5), and argue in a manner similar to (3.3.6) and (3.3.7) to obtain ~r6((x) 3p(x) 6(1 x2)2n3(n 1)3P31 ) = O n4 v=2 1si sn sin t, sin Ll = 0 (n2) n 1 n1 ( )P4 P,2E _, (xk)i,.(xk)Ek(x) =2 (1 2 (2 ) k=2 ( IE(x 1 ) (n2) (1 x2D sin t, sin This completes the proof of the lemma. 3.4 Estimate of the Fundamental Polynomials C,(x) and C,(x) Lemma 3.4.1 We have IC,(x)l = O(n4), ICn(x)l = (n4). ssi tk k sin ) 2 =0 (4 k=2 (n1 k\=2 =nO kc=2 Proof Using (2.1.3), (2.1.8), (2.1.11) and (3.1.5) we obtain (1 x)xr(z)() (1 x)~rx(z) 4n(n 1) 4n(n 1)(: 1)27r(1)2 4n3(n 1)3(1 = n Using (2.1.8), (2.1.11), (3.1.5), Markov's and Bernstein's inequalities 2n2(n 1)2 2nn 1( )2(1)2 2n2(n 1)2(x 1)27r,(1)2 =0 (1 ) ()PPI(x)rn(x) S 2n4(n 1)(x 1)2 ((1 + x)2n (1 2)n (n4), 2n4(n 1)4 and finally, using (2.1.9), (2.1.3), (3.1.3), IPni(x)l < 1 and Lemma 3.2.2 n1 12 z Pn(l(xk) k)ex2(1 k=2 +0 (n(n 1) S n1l PI(k) E( ) Sk=2 +0 (n(n n(n 1)( 2 (zk  1))Ek(x) n1 Xk 1)P2_1(xk) (xk)Ek(x) k=2 n1 4 L 1) 4 )1 k=2 n1=2 k=2 P"i(kX k)2Ek(x)) Ek(x) n(n 1) n 1 1 Ek() (= 1 k2k2 2( ) X2)2 Xk k=2 k=2 = ( Ek(X) =(n'). _(1 X2)2 k=2 k Noting that C,(x) = Ci(x) completes the proof. 3.5 Estimate of the Fundamental Polynomials of the Second Kind Lemma 3.5.1 We have (3.5.1) IB(x) = O(n2), (3.5.2) n1 B IB( x)\ (1 ( 2) v=2 v I Proof We show first Bi(x)l = O(n2). inequality and ef(x) < 1 we obtain 7,(X~) (X) 7 { (x)1(() = (1 + X)2)(1  Sn2(n 1)21 Applying (2.1.8), (2.1.11), (2.1.1), Bernstein's S (1 x 2)pl(x) (x)2(x)) (x 1)n2(n 1)2 )p._l()2) = O(n2). Next, applying Markov's inequality (to e (xk)), (2.1.8), (2.1.3), (3.1.3), IPnl(x) < 1 and Lemma 3.2.2 yield 24 1 24 nI 7r(Zk)e (Xk)3Ek(X) n(n 1) k=2 = ( (= X (k) Xk)EkX) k=2 IB(x)I = O(n2) 0(log n) O ( n21 Pn 1( k) k=2 (1 xk)2 k ) =0 ( n2 (Ek()2 = O(n2) k=2 k );/ Observing Bn(x) = Bi(x), we deduce that (3.5.1) holds. Using (2.1.1), (3.1.9), (3.1.4), (3.2.17) and l3(x)i < 1, we note that for It t.\ < n nr(x) (x) (1 x)in(n 1)P,_ (x,) S)Z 2 pn,1 ^^/(l^)n P (1V)=4n2(n) S0(n1). Observing (2.3.11), (3.3.2) and (3.1.7)(3.1.8) we have n1 S4n(n 1) I( v=2 1X n1 0 (2 E v=2 n1 n 13n(n 1)x, E=(1 X)"(1 X2)2 (log n Vn IE,(x)\ (1), = O (n3 n1 E_.(x) >= ( n1(1 X2)2 Now, applying (2.1.3), (2.1.8), (3.1.5), (3.1.9), (3.1.11) and e2(x) < 1 we obtain S7rn(x)(x) It (1  x)2n(n 1)Pn_1(x,) 0 sin tn2(n 1)2P,21(X,)_  \tt ,l> (n? n1 n 21 I D.(x)\\ Sv=21z 2 1 =22 sin t (n 1)2X xr o nY, =o(n 2 I t'i>i SO log n) \ n sin sin h2 ) and further using (2.1.9) and (3.1.3) yield n1 24 n1 24 Ev2 (1 2) k=2 Ek(x) (1 ~)2 ItktLl> 7rn(k k) Ek(2) P'L(Xk)(1 XD)2 (1 x,)Llxk X3P, ,) k2 22_,(X,) k 3 41i ( IEk(x) k tkt\l> n 1 0 (n3E \ k=2 IEk(x) ( k ) 1i ) sin3 2 = O(n). This completes the proof of the lemma. 3.6 Estimate of the Fundamental Polynomials of the First Kind Lemma 3.6.1 We have SIA~(x)l = 0(1). v=1 Proof We show first that IAi(x)l = 0(1). It suffices to note that on using Bernstein's inequality, it (x)I < 1 and Lemma 3.2.2 IEk ) = (1), (1 Zk2/' n1 n1 S(Xk)' E,(x) = O n4 k=2 k=2 El> G*I>{ = It (n1 k=2 kf=2 S(n k=2 46 and observing A,(x) = A,(x), we deduce IA,(x) = 0(1). Now, as fe(x) < e~(x), we have n1 Ze4 (x) < v/=2 n1 3 ,e(x)) <1 v=2 and observing (3.3.2), (3.1.7)(3.1.8) and Lemma 3.2.2 we have ( 0 n12 IDV(x)I S log n) n IE,(x) (1 zx)2 n1 E \x1)3 n1 S1 E, (x) v=2 (1 z)= ) =0(1). It remains only to estimate n1 n1 S( 24 Z" (Xk)4Ek(x) v=2 v=2 and applying (2.1.9), (2.1.3), (3.1.3), (3.1.9) and (3.1.11) yields (n (IEk(X) I \E(1 x2)2 c k=2 Itkt l> n ( Ek( Ik(1 X tlI)2 \k=2 Itht l: (1 x2)2p,(xk) (Xk X )4P4_ 1(x) c k x,4 4n(n 1)x, v=12  v=2 " S24n(n 1) v=2 5(1 x2)2 v.=2V Sn1 =0 n E v=2 n2 n + 3 Ex(2) ) \ 1 4 = 0 (n4 = (I x ) sin2 tk sin2 t, \k=2 2 I tktI> 2 \k=2 tk s> n 2 =0 _k(x) = 0O(1). Sk= (1 k) This completes the proof of the lemma. O 3.7 Proof of the Convergence Theorem Denote by [x] the largest integer less than or equal to x. This expression is known as the greatest integer in x. Let f C[1, 1], m = and consider polynomials pm(x) of degree at most m such that (3.7.1) 1 IIf/)(z) p )(2X)I = O(m()w4(f,  m and (3.7.2) IIP$)(x)II = O(m()wj(f, ), 1 < j 4 m The above polynomials exist by a paper of H. Gonska [16], page 165. Since 'modified' (0, 1,3,4) interpolation is uniquely determined, we have n n pn(x)Rn(Pm, x) = Ep',(x,)B,(x)+p (1)Ci(x)+p(l)Cn(x)+E .(x.)D,(x) v=1 v=1 n1 + Ep)(x.)E,(x). v=2 Thus, using (3.7.2) and Lemmas 3.5.1, 3.4.1, 3.3.1 and 3.2.2 we have n n IPm(x)R(pm, x)l = Zp'(x,)B,(x)+p (1)Ci(x)+p"(l)Cn(x)+L p'"(x,)D,(x) v=l 1L=l ni1 + E PM)(xv,)E(Zx) v=2 (3.7.3) n1 = o(m)wi(f, 1) IB.(x) + O(m')w2(f, )(ICi(x)l + IC(x)I) v=1 n n1 +o(m3)w3(f, lD.() + O(m )w4(f, 1) .(2)l. V=1 v=2 Now, from (3.7.1), (3.7.3) and Lemma 3.6.1 we get Ilf Rn(f)II < Il PmII + 1pm Rn(pm)I + IIRn(pm f)ll 5 Ilf Pmll + Ilpm Rn(pm)II + ip,  + 0w (f, log n +o^(/,^n fll II E IA,(x) II uV=1 + 0 (W4f, ) n/.  (f logn  This proves the theorem. W4 log n = 0 (wj CHAPTER 4 ERDOSTYPE INEQUALITIES 4.1 Main Results Let us denote by S, the set of all polynomials whose degree is n and whose zeros are all real and lie outside (1, 1), and denote by L, the set of all polynomials of the form (4.1.1) n Pn(x) = akqnk(x), ak >O (k =0,1,...,n) k=O where qnk(x) = (1 + x)nk(l X)k Here we present two theorems concerned with finding a uniform upper bound for the expression S1 w(x)(Pn(x))'dx 1, w(x)(P,(x))Pdz where w(x) = (1 xa2)a,a > 1, when p = 2, and w(x) = (1 x2)3, when p = 4, and where the polynomials P, are restricted to the Lorentz class Ln of polynomials. It is known [22] that if P, E Sn, then P, E Ln or Pn E Ln. Thus, Theorem 4.1 is an extension of the classical theorem of P. Erd6s [10] for Pn in Sn, as well as the results of P. ErdSs and A.K. Varma [11] and the Theorem 3.4 in Milovanovi6 and Petkovi6 [28], in the L2 norm with the ultraspherical weight w(x) = (1x 2), a > 1. In Theorem 4.2 we present the first sharp extension of the inequality of Erdos in a weighted L4 norm. Note that Theorems 4.1 and 4.2 provide the polynomials which attain the given upper bounds. We shall later see that for each n there exists a unique positive solution to the equation 24 + (8n5)a3 +(12n2 17n+4)a2 +(8n3O 20n2+lln 1)a2n(2n2 5n+4) 0. Denote this solution by an. Theorem 4.1 Let Pn E L,, n > 2 and a > 1 real. Then we have for a > an (1 x2)a(P'(x))2dx < 2(2n + 2 n + (1 x2)'(P())d 12(2n + a)(2n + a 1) with equality for Pn(x) = c(1 z)n. For 1 < a < an the inequality becomes j(1 )P,, ))2d <(2n + 2a + 1)(n + a)[a(a 1)n2 + 2(n 1)(n (a 1)(2a 1))] 2(a + 1)(a + 2)(2n + a 2)(2n + a 3) x (1 x2)(P 2dx 1 with equality for P,(x) = c(1 x)nl(1 t x). Theorem 4.2 Let Pn E Ln. Then we have f' 21 I < n3(4n + 7)(4n + 6)(4n + 5)(4n + 4) n( 64(4n + 3)(4n + 2)(4n + 1) with equality if and only if P,(x) = c(1 x)". 4.2 Some Lemmas Lemma 4.2.1 Let P, E L, and a > 0 real. Then we have (4.2.1) (x))2 dx <(2n + 2a + 1)(n+ a) (1 1 2a(2n + a 1 S(1 x2)3(P,(x))4dx _z ^^)4 x ), P 2))2dx with equality if and only if P,(x) = c(1 x)". In the case P,, L,. and P,(1) = 0, the inequality becomes (4.2.2) (1 2a <(2n + 2a + 1)(n + a) (I_ 1 2)1(P(X))22dx (4.2.2) (1 x2)'i(P,(x))2dx < 2 )2+ (l 2)a(Px))d 11 2(a + 2)(2n + a 2) 1 with equality if and only if P,(x) = c(1 x)n'(1 F x). Proof We write Q2n(x) =(Pn())2. Then Q2n E L2n and we have from (4.1.1) 2n Q2(x) = ak(1+ )2nk(l . k=O Thus, (4.2.3) f~,(1 x2)_l(P,(x))2dx ko ak fl(1 + )2n+alk(1 _ )k+ldx _1(1 x2)a(P,(x))2dz o ak (1 + x)2"+ck(l x)k+aod and we use the known formula (4.2.4) IF(p + 1)F(q + 1)2p++ 1 F(p + q + 2) to obtain fj(1 + x)2n+lk(l )k+aldx r(2n+a)r(k+a)22"+2a __l_____ ____________ ;_ Pr(2n+2a) f(1 + x)2n+.k(1 x)k+adx r(2n+a+lk)r(k+a+1)22n+2a+ 1 rF(2n+2a+2) S(2n + 2a + 1)(2n + 2a) (2n + 2a + 1)(n + a) 4(k + a)(2n + a k) 2a(2n + a) for k = 0,1,2,..., 2n, where equality holds iff k = 0 or k = 2n. Applying the above to (4.2.3) yields (4.2.1). If P,(+1) = 0, then we note that k runs only from 2 to 2n 2 above. Employing this observation in (4.2.3) proves (4.2.2). This completes the proof of the lemma. O Lemma 4.2.2 Let P, E L, and a > 1 real. Then we have (4.2.5) (1 x2, p dx < an (1 X2),(Pn(x))2d J1 2n + a 1  with equality if and only if P,(x) = c(1 4 x)". In the case that P, E Ln and P,(1) = 0, we have for a > 1 real the inequality (4.2.6) (1i 2)( ))d < a(a 1)n2 + 2(n 1)(n (a 1)(2a 1)) (a + 1)(2n + a 3) x (1 x)(P,(x))dx with equality if and only if Pn(x) = c(1 z)nl(1 T x). Proof From (4.1.1) we have (P(x))2 = E j=0 n E akajqnk(x)qnj(x) k=0 so that we may write  x2)a'(P(x))2dx n n= = E aka= j=0 k=0 = akaj ( + )2+a1k( x)k+j+1 dx. j=o k=O 1 On using (4.2.4) and writing = k + j we obtain (4.2.7) j(1 X2)l(P(X))2d" akaFj(2n + a )r( + a)22n+2a' 1(1 '( )dx = = k= F(2n + 2a) (1 1 2 (1 x qk()qnj()dx 1 Next we show that for a > 1 (1 x2)(P(x))2dx = E aka (1 X)qk(X)q(x)dx 1 j=0 k=0 1 (4.2.8) < an 2 akajr(2n + a )r(y + a)22n+201 2n + a 1 F(2n + 2a) j=O k=O As qnk(x) = (1 + )k)(1 ) we have qnk(x) = (n k)(1 + x)nk )k k(1 )nk(l )k1 and using (4.2.4) yields (4.2.9) [1 22n+21 r SIk 2n d+2cr [(nk)(nj)r(2n+a1)F(t+a+1) Ikj 11 )'qnk(x)qnj(x)dx = r(2n + 2a)  +kjr(2n + a + 1 )r( + a 1) (nt 2kj)r(2n + a )ry( + a)] r(2n + a )r(e + a)22n+2a1 r(2n + 2a) 2n+a1 +a(2n+2k We denote the portion in brackets by pkj and simplify the expression as follows, denoting e = k + j and later using that 4kj = 2 (k j)2. We have (n2 ni + kj)( + a) kj(2n + a i) 2kj + nI + 2kj k= 2n+al1 +( 1 an2 + n2t n(2n + 2a 1) kj(2n + 2a 1)(2n + 2a 2) 2n+a1 2n+al1 + (2n+a )( + a 1) an2 [2n2 + (2n e)(2a2 3a + 1 n)] 2n+a1e 2(2n+a 1)( +a1) (k j)2(2n + 2a l)(n + a 1) 2(2n +a 1 e)(e+a 1) 2a(a 1)n2 + (2n )(n (a 1)(2a 1)) (k j)(2n + 2a 1)(n + a 1) 2(2n + a 1 )( + a 1) We show that Iki > 0 for k,j = 0,1,.., n, (a > 1) and apply this to (4.2.9) to obtain (4.2.8). We have an2 an2 2n+ a  1 k = 2n + a 1 2a(a 1)n2 + (2n )(n (a 1)(2a 1)) (k j)2(2n + 2a 1)(n + a 1) 2(2n+a 1 )(+ a 1) > e(2n )[2(a 1)n2 + (4a 3)(a 1)n + (a 1)2(2a 1)] > 0 2(2n +a1)(2n+a 1 )(+ a1) 1 with equality iff f = 0 or t = 2n. Observing (4.2.8) and (4.2.7) we have (4.2.5). Lastly, for the case P,(l) = 0 (k,j = 1,2,...,n 1) and 1 < a < 1 we have a(a l)n2 + 2(n l)(n (a 1)(2a 1)) (a+ 1)(2n+a3)  kj (2n 2(n 1)a a2)[f(2n ) 4(n 1)] (a+ 1)(2n + a 3)( + a 1)(2n + a 1 ) (k j)2(2n +2a 1)(n+ a 1) 0 (f+ a1)(2n + a1 ) as 2n 2(n 1)a a2 > 1 for 1 < a < 1, and (2n t) 4(n 1) > 0, with equality iff k = j = 1 or k = j = n 1. This yields (1 x2)"(P'(x))2dx a(a 1)n2 + 2(n 1)(n (a 1)(2a 1)) (a + 1)(2n + a 3) S akajr(2n + a )r( + a)22n+2a1 j=0 k=0 r(2n + 2a) Combining the above with (4.2.7) yields (4.2.6), completing the proof of Lemma 4.2.2. O 4.3 Proofs of Theorems Proof of Theorem 4.1 Let Pn E L,. The case a > 1 follows from (4.2.5) and (4.2.1). Now, let n > 2 and 1 P,(x) = ao(1 + x)n + Q,(x) + an(1 x)" where Q, E L, and Qn(1) = 0. Then (4.3.1) f(1 x2)(P(x))2dx = n2a2 1i J (1 x2)r(1 + x)2n2dx 1 +n2a 2 (1x2 )2n2dx+ ( 1 x )a(Qn(x))2dx 21 1 2n2aoan (1 x2)n+ldx + 2nao (1 x2)(1 + x) Q'Q(x)dx 1 J1 2nan (1 x2)a(l x)b'Q'(x)dx. We show the last 2 integrals are nonpositive. Integrating by parts we obtain II = 2nao (1 2 )'(1 + )'Q'(x)dx = 2nao (1 x)a(l + x)"+lQ',(x)dx J1 1 = 2n(n + a 1)ao (1 + x)"+(2(1 _ X)Qn(x)da J1 1 +2naao 1 (1 + X)"+1(1 x)alQ.(x)dx. J1 Thus, we have II < 0 for 1 < a < 0, and if for 0 < a < 1 we show (4.3.2) S1,(1 + x)n+al(1 )alQ.(x)dx 2n + a 2 +(1+ Z)" (  z)Q(x)dx 2a then we will have fl I, 5 2n(n + a 1)ao (1 + x)n+"2(1 x)Qn(x)dz +(2n + a 2)nao (1 + X)n+2(1 z)oQn(x)dX r1 aaon I (1 X)a(l + x)n+o2Qn(x)dx < 0, J1 as desired. We show now (4.3.2). It suffices to consider S(1 + x)n+lx(1 x)''qnk(x)dx f,(1 + x)n+O2(1 x)aqnk(x)dx f 1(1 + )2n+a1k(l X)k+aldx 2n + a 1 k f_1(1 + X)2n"+"2k(1 X)k+fa k + a 2n+a2 < + (k =l 12, ...n 1) +l 2n +a 2 2a (0 < a < 1) showing (4.3.2). In the same manner we obtain 2 = 2na (I x2)a(1 )"'Qx1(x)dx = 2na, (1x)n+('(1+Xz)Q',()dx 1 J1 = 2n(n + a 1)a,, (1 + x)*(1 x)n+"2Qn(x)dx J1 1 +2naan (1 + x)'(1 X)"n+'Q.(x)ds, J1 showing that I2 < 0 for 1 < a < 1. So from the above, (4.3.1) and (4.2.4) we obtain (I1 _r 2)((2n + a 1)r(a + 1)22n+2a' /(1 nn2) F(2n + 2a) + j(I z)(Q ))2d J1 and noting that 1~ 2)(P r(2n + a + 1)r(a + 1)22n"+c+1 0 n rF(2n + 2a + 2) + (1 X2)(Qn(x))2dx J11 yields (1 x2)Y(p'(x))2dx f_(1 x2)(P,(x))2dx (4.3.3) n2 ) F(2n+a )r(a+1)22n+2a1 ))2 n2(ao + a )(2na+l +22 + ,(1 z2)a(Q,(x))2dx Now consider + a)(Q )d Now consider f(1 x12)(qo(X))2dx _, (1 x2)l(qno(X))2dx f,(1 x2)(ql(x))2dx ,1(1 x2)a(qnl(x))2dx (n 1)(2n + 2a + 1)(n + a)f,(a) (a + 1)(a + 2)(2n + a)(2n + a 1)(2n + a 2)(2n + a 3)' where fn(a) = 2a4+(8n5)a3+(12n217n+4)a2 +(8n320n2+lln1)a2n(2n25n+4). As the denominator of the above ratio does not change sign, we need only look at fn(a). It not difficult to check that f,(a) is increasing on (1, oo), and that it has precisely one positive zero, which lies inside (0, 1). We call this zero an. From above, we see that for 1 < a < a, n2(2n + 2a + 1)(n + a) <(1 x2)*(q~o(x))2dx fl(1 x2)a(q())2dx 2(2n + a)(2n + a 1) f 1(1 x2)a(qo(x))2dx f(1 x2)c(qn,(x))2dx (2n + 2a + 1)(n + a)[a(a 1)n2 + 2(n 1)(n (a 1)(2a 1))] 2(a + 1)(a + 2)(2n + a 2)(2n + a 3) and for a > a, the above inequality is reversed. Employing this observation in (4.3.3) completes the proof of Theorem 1. O For the Proof of Theorem 4.2 we shall use the inequality for P. E Ln and 1 < x < 1 (4.3.4) (1 x2)((PF(x))2 P,(x)P"'(x)) < n(P,(x))2 2xP,(x)P,(x). This inequality is found in the paper of MilovanoviC and PetkoviC [28], page 284. Proof of Theorem 4.2 Multiplying (4.3.4) through by (1 x2)2(P,(x))2 and (1 x2)(Pn(x))2 we obtain the inequalities (1 x2)3I(P(x))2((P,(x))2 Pn(x)P'"(x)) (4.3.5) < n(1 x2)2(P,(x))2(P, ())2 22x(1 x2)2P()(P'(x)3 and (4.3.6) (1x2)2 (Pn())2((P:(x))2Pn(n)P2())n < n(l 2)(Pn(x))42x(1x2)(P.(x))3P,(xt). Denote = (1 x2)3(P(x))4dx. 1 Integrating by parts yields 1 1 I= 3 j1 (1 X2)3(p(x ))2P.(X)P,'(X)dx + 6 j x( X2)2(P, (x))3P,(x)dx, and adding 311 to both sides we obtain 411 = 3 (1 (()2((,(x))2P"(x)P(x))dx+6 (1x2)2(p(x))3Pn(x)dx. 1 1 Applying (4.3.5) to the above yields the inequality (4.3.7) 4 (1 x)((x))dx < 3n (1 P(x))((dx. 1 1 Now, for any polynomial P,(x) we have d 4(1 x2)2(Pn(x))2(P()2 )2 [(1 x2)2(p(s))3p(x)] = (1 x2)2(P,(x))2(P(x)2 + 4x(1 x2)(P,(x))3P,'() (1 x2)2(P,P("(). Thus, integrating both sides of the above from 1 to 1, and then applying (4.3.6) we obtain 4 f(1 x2)2(P,())2(P,(x))2dx = JI (1 x2)2(P,(x))2((P'(x))2 Pn(x)P.(x))dx 1 +4 x(1 x2)(P,(x))3pn(x)dx Il 1l 11 1 Integrating by parts the last term above yields the inequality 4 (1 x2)2((Pn ))2(p :())2dx < (n ) 1 2 f, ( (1 x2)(P(x))4dx + (Pn(x))4 dx. 1 Applying the above to (4.3.7) we get ( x)(P,(x))4dx < 3n(2n 3) 1 (1 2)(P,(x))4dz + 6 f(P())dx. Now, replacing n by 2n in (4.2.1) with a = 1, and then a = 2 and a = 3 yields l(1z2)3(P,(x))4dx < J 1 3n(2n 3) 32 3n(4n + 3)(4n + 2)) /1 64(4n + 1) =2n (1 x)(P,(x))4dx 2(4n + 1) 1 n3(4n + 7)(4n + 6)(4n + 5)(4n + 4) (1 )(P.))'d  64(4n + 3)(4n + 2)(4n + 1) J_ completing the proof of Theorem 4.2. (1x')(Pn( x))dx CHAPTER 5 TURANTYPE INEQUALITIES 5.1 Main Results Let Hn denote the set of all polynomials of degree n whose zeros are all real and lie inside [1, 1]. Inspired by the inequality of Turin [45], we present three theorems concerned with finding a uniform lower bound for the expression 1, w(x)(P'(x))Pdx w (x)(P ,(x))Pdx for P, E Hn. We provide sharp inequalities for the special cases p = 2 and w(x) = (1 x2), a > 1 (a > 1 if P,(l) = 0), and p = 4, w(x) = (1 x2)3. Then we present an asymptotically sharp result for w(x) 1 and p even. Theorem 5.1 generalizes some previous results of A. K. Varma [51],[50], and Theorem 5.2 extends these in a weighted L4 norm. Theorem 5.1 Let P, E H, ,n > 2 and a > 1 real. Then we have (n = 2m) (1 n2(2n + 22(p)) + 1)2 J (1 x2)a(P,(x))2dx 1 4(n + a 1)(n + a) 1 with equality if and only if P,(x) = c(1 x2)m. If P,(1) = P,(1) = 0, then the above remains valid for a > 1. Theorem 5.2 Let P, E H,. Then we have (n = 2m) 1 x p 3n3(4n + 7)(4n + 5) > (1 )(()) d 1( x)3(P(x))4dx 4(4n + 6)(4n + 4)(4n + 2) (1 with equality if and only if P,(x) = c(1 x2)m Corollary 5.2.1 Let Pn E Hn. Then we have (5.1.1) (1 )(P(x ))dx> 8) (1 x2)(P,())4d >1 8(2n + 1) _1 and (5.1.2) I 3n3(4n + 5) (P()) (1 x2)2(P())4d > 32(n + 1)(2n + 1) _ where these results are the best in the sense that there exists a polynomial P*(x) of degree n having all zeros inside [1,1] (P,(x) = (1 x2)m,n = 2m) and for which and f,(1 x')(P,*'())4dx f(1 x2)2(p,'(x))4dx Sf,(1 xz)2(P*(x))4dx f(1 I2 2P.*(X))4dz 3n3(4n + 3)(4n + 1) 64(2n + 1)(2n 1)(n 1) 3n3(4n + 5)(4n + 3) 64(n + 1)(2n + 1)(2n 1)' Theorem 5.3 Let Pn E Hn and p > 2 even. Then we have ((p l)n (p 2))((p 3)n (p 4)) (5n 4)(3n 2)n(pn + 1) p (pn 2) where this inequality is sharp in the sense that for the polynomial P*(x) = (1 x2)" (n = 2m) we have 1 IIP,*'I (p1)(p3)...53 lim L noo n2 IIPnl p2 (Pn(x))Pdz J (P'(x))Pdx > 64 5.2 Some Identities We shall need the following known identities. (5.2.1) 1 P (x) = P (x)Z 1 X Xk k=l (5.2.2) (P()) P,(x)P,'(x) = (P,(x))2 1 ( k=1 (Z Xk) (5.2.3) 1 2 2x 1 z +  = 1+ (x xk)2 x Xk ( Xk)2 Identity 5.2.1 Let a > 0 and P,(x) be any algebraic polynomial of degree n with x1, X2,... xn as its real zeros. Then we have 2 (1 x2)a+l(P(x))2dx = (n (2a + 1)a) (1 X2),(p))2x 1 1 +2a2 (1 x IP())dx 1 1 + (1 2)(P_,(x))2 2 d. (1 k=1 k)2 1 k=l Moreover, if P,(x) vanishes at x = 1 and x = 1, then the above is also valid for 2 < a < 0. Proof Integrating by parts, we have for a > 0 (1 2)a+l p ()2dx = f P,(x) [(1 x2)a+lPP,(x) 2(a + 1)(1 x2axP'(x)1 dx. 11 Therefore, we obtain 1 1 2 j(1 x2 (p x)2dx = [(P,(x))2 P(X)P:,(x)] (1 x2)+dx +2(1 + a) (1 x2)a xP(X)P,(x)dx. Now, on using (5.2.1)(5.2.3) we have 2 (1 x2)+l(p,(x))2d = (1 2)(P_(x))2 E l d 1 k=1 k) + 2(X2) (Pn()x dxa j(P,(x))2{(1 2) 2ax2(1 ad k=1 k 1 1  S (1 2)(Pn,())2 dx + (n a) (1 X2)"(P,(x))2dx J 1 (X k)2  2a2 a2 ( 1)(1 x2)P ))2d ( 1 x) d2 (1 X2)(pn(X)) (x xk)2 d 1 k=1 k) +(n a(2a + 1)) (1 x)=(P,+(x)) dx 1 +2a' (1 2 )aX(P(x)) 2dX. This proves the identity for a > 0. For the case Pn(l) = 0 and 2 < a < 0, the above proof remains valid. Thus, Identity 5.2.1 is established. O Identity 5.2.2 Let Pn,() be any algebraic polynomial of degree n with n real zeros x1,x2... x,, and let a > 0. Then we have ( X2)(pn(X))2dx (2n + 2 + 1) pn dx 1 (2n + 2a) _ 1 1 1 x Sdx (1 ')(Pe(x))' 1 k dz 2(n + a)2Jl 2 ( Xk)2 + ( 1 1 (1 x2)a1[(1 x2)P,(x) + nxPn(x)]2dx. (n + a)2 I Moreover, if P,(1) = 0, P,(1) = 0, then the above is valid for 2 < a < 0 as well. Proof First we note that for a > 0 +n2 j( x2)1'x2(P(x))2dx + 2n (1 ')oxP.(x)P;(x)dx 1 1 ji [ x2 P,)dx + n2 f x2(xXP))2dx 1 1 i n J(P+(X))2[(1 X2)d 2X2a(1 X2)al]dx j x2)+l(P()x))dx + (n2 + 2na) x(1 X))2dx n J (P,(x))2(1 x2)dx 1 1 = (1a + )a) (P '( p(X))dd (+2n ) 1 (1 Z2)a(Px (X))2dz 21 1 +(n 2+2n) 2 (P(x))( 1 2)'dx = ( 1 ) ( )) (P2 +2 z(1 )2 a1 r( 2dx 2 11 nPn + 11 ))2 1 1 12 + (1 (P(x) 2 dx 2 k=1 (x Xk) (n2 + 2na + n) (1 x)(P,())2dx 1J +(n2 + 2na) (P,(x))2(1 x2),1dx 1J1 (1 P ) 1(Pi dx + jni(2+ 1 (n + 2na + n))P (1 X2,(P. jdx + (1 )(P())2 k 2dx , 1 k=l which is equivalent to the stated identity. We note that if 2 < a, then the identity is still valid provided P,(1) = 0, Pn(l) = 0. This proves the Identity 5.2.2. O Identity 5.2.3 Let Pn be any algebraic polynomial of degree n with X1, X2,.. *n its real zeros. Then we have (1 X)(P())4dz = 3n (1 x2)2(pn X))2(p'(X))2ds f(l x p(x))dx = f dx 1 4 1 3 r 1 2 . + (1 X2)2(pn)2())2 E ( Xk d. 4 k=1 X k) Proof Denote J= (12)3(p 3 ))4dX. J1 Integrating by parts yields I = 3 (1 x2)3(:(x))2P,(x)P,(x)dx + 6 (1 x)(()) (x)dx and adding 31I to both sides we obtain 41I = 3 (1 x2)3(P.(x))2 [(P,(x))2 P,(x)P'(x)] dx +6 x(1 x xy)i(Pn(x3P(x)dx. Applying (5.2.1(5.2.3 yields the identity. Applying (5.2.1)(5.2.3) yields the identity. Identity 5.2.4. Let Pn be a polynomial of degree n with real zeros x1, x2,... ,X. Then for p > 2 even, we have (1 1 (p r 2) 1 (P )(P )dx + (r + 2) 1 (1 (P ))))' Ji Ji = ((pr 1)n (pr2)) Jl(P,(X))pr2(P(X))r+2dx J11 +(p r 1) ))+2 x k) 2d k1 =1 x +(p r 2) (P,(X))p"2(Pn(x))'(xP,() Pn(x))2dx, J1 r = 0,2,...,p 4. Proof Denote I, = (1 x2)(P())P'(P(X))rdx. 111 Integrating by parts we get Ir += 1 r (1 X')(P,(x'))(r'P x()P )P."(dx r+1 j1 r + 1 _ and adding plIr to both sides yields P I =Prl1 rI1 r=_ (1 x2)(P'(x))p '2(P ,())((P'(x))2 P,(x)P'"(x))dx r +1 r+1 JX P rS 1 j(P^(X))P'r(P,(X))r+1dX 2 r1 1 +2 (P,'(x))rl(pn(x))r+ldzx r + 1 1 +( 2( r )) x( P(x))prl(Pn(x))r+ldx. + r+l r+l 1 We now multiply both sides by r + 1 and apply (5.2.1)(5.2.3) to obtain pIr = (p r )n (P'(x))P2(P, )r+2d 1 +(p r 1) f(P(x))r(P(x))r+2 E 2d 1 k= (x xk)2dx 2(p r 1) x(P.,(x))Prl(P,(x))r+Idx (p 2) j (P(x))P (P(X))(XZP'(X)) P d(x))2dx J1 +(p r 2) (P,'()),r2(P,(X))r(XP,'() P.(X))2dx J1 = ((p r 1)n (p r 2)) (P'(x))P(P(x))r+2d +1 (Xn)r+2 dx +(p r 1) (Pj'(x))pS2(P1(x)) + A d 1 k=1 Xk) k=1 +(p r 2) (P,(X))p 2(P,()) (XP,(X) P.(x))2dx 1 (p r 2) 2 x 2 X))pPn )rdx, 11 and taking the last term above to the left hand side completes the proof of the identity. O 5.3 Proofs of Theorems Proof of Theorem 5.1 On using Identities 5.2.1 and 5.2.2, we obtain J1 1 1 12 +22 [(2n + 2a)c +1 ) (1 _(2(pr(l))2) J (2n + a2) ) 1 Xk2 dx /(1 z) (P,())2 k 2( dz 2(n +a)2 1 k= k) + ) (1 )' [(1 x2)P(x) + nxP,()]2dx + (n + 2n)  [ + (1 _2 )(PX))2 k 2d x na (2n + 2a + 1) =+ (1 X2 ) .(x2)) (P(x d)) ddx (n + a)2 1 n2 1 n2 + 2na 1 1 x)2 = 1 (1 X2),(Pn(x))2dx + (n + j)2 I_ 2 E k dx n + a 1 k=1 k) 202 F1 (n+(1 )"'[(l x2)P(x) + nxP.(x)]2dx (n + a)2  n2 (2n + 2a + 3) 1 (1 _2)o+'(P.(X))2d n+a 1(2n + 2a+2) 1 1 1 12 X (1 $2) +l(p1n())2 k (5 dz)2 2(n + a + 1)2 1 )k= )2 k= 1 + 1 (1 x2)[(1 x2)P,(x) + nxP,(x)]2dx] (n + a + 1)2  n2 + 2na 1 1  + 2 (1 x2)a(P(x))2 E 1 k )2 +(n + )2 1 k= ( k 292 f 2 + 1 (1 x [(1 x2)P(x) + nxPn(x)]2dx. (n + a)2  Note that Therefore, we have J ) (n + a)(2n + 2a + 2) J1 proving the theorem. O Proof of Theorem 5.2 Let P,(x) be any algebraic polynomial of degree n with xl, 2,..* as its real zeros. Then we have 4(1 x2)2( (x)2P)2)(n 2 ( [(1 x2)(P,(x))3P,(x)] fl (n +a)(2 + 2 + 2 = (1 x2)2(P,(x))2(P,:(x))2 + 4x(1 x2)(P,(x))3P(x) (1 x2(P,())"P(). We integrate both sides from 1 to 1, and make use of (5.2.1)(5.2.3). We then obtain (5.3.1) 4 (1 x2(Pd(x ))2 = (1 )2(P(x))2 ))2 P.(x)P,(x))dx +4 x(l x2)(PF(x))3P,(x)dx 11 (1 /'1  x2)(Px))4 1 2 + 2x(x xk)dx + 21 (1 x2)(P,(x))3~(x)dx k=1 ( k )2 J = f(1 x)(P(x))4 k dx + n (1 x2)(P,(x))4dx Jl k=L Jl 1 /1(1 3x2)(p,(x))4dx 2 1 1n 1 _X2 3)1 1 = l(x2)(P(nx))4 k dx+(n) I (1x2)(Pn(x))4dx+J (P,(x))4dx. 1 k=l ) 1 J From Identity 5.2.3 we have S(1 x2)3(P,(x))4d n (1 2)2 )) 2 p )) 2 JI 4  + (1 x2)2(P(x))2(P(x))2 (X d 4 1 k=l k) Now, using (5.3.1) we obtain (5.3.2) 1(1 x2)3(p(x))4dx = n [ ( fl 16 3) x( 'x)(Pn(x))dx + f(Pn())4dx 2/ 11J + (1 x2 (P )4 k 2 1 k=1 k) +3 (1 x2)2(p,())2(P.(x))2 4 1 k=l 1zx x  dx. (X Xk)2 Also, from Identity 5.2.2 with a = 1 and n replaced by 2n, and on noting that in this case 1 2 (x Xk)2 we have n ?1 =2E k=1 1 2 (x Xk)2 (P(x))d > (4n + 3) (1 2)(P(x))4dx 1 1x d (1 x2)(Pn(x))4 Xk dx. (2n + 1)2 1 k k Therefore, on using (5.3.2) and the above we have (5.3.3) J(1 x23(Pxdx > 3 (1 2)(P 2) (x))4 dx 2 1 4n+ 3 fld +2(2 ) 1 (1 x2)(Pn(x))4dx 2T(2n + 1)_I 1 n, 1 X2 ] j (1 x2)(Pn(x))4 k 2kdx 1 k=1 x k)2 3 .1 (1 x )dx + (1 x )2P (x))IJx 2 (x xk )2 1 k=1 [2n2 (1 2)(pn(n + 1) J 1)) 2n + 1 (2n 1)2 1 k=1 3 1 3 22p(p\2p'nX )2) 1 xk dx 4 (1 2)2 Xk)2d. 4 1 k=1 3n 16 1(2 ) (2n + 1)2 1 X2 k)i~ Again, using Identity 5.2.2 with n replaced by 2n and a = 2 we have (5.3.4) (4n + 5) / (l x2)2(P,(x))4dx 1(1 x)(P.(x))dx > (n + ) (1 dx 1(4n + 4) 1 (1 2j(i x2)2(Pn(x))4 1( k)dx )2 ( ( 2Pk )2 dx, (2n + 2)2 1 k=1 and similarly with a = 3 (5.3.5) l (P())dx (4n+ (1 2)(P(x))4dx 1 1 u  1 x 21 1 (1 x 2)3(pn(x))4 1 k 2dx (2n + 3)2 1 k=1( X k)d Therefore, we obtain (1 2)(Pn())dx > (1 P2)(n())4dx 4n + 5 4n + 7 4n + 4 4n + 6 _1 1 (1 (2n+3)2 1 1 x1 dx] k=1 (X Xk)2 1 1 x 2 2 (1 i x2)2(P~(x))4 Z ( )2 dx (2n + 2)2 _1 k=1 (X k) (4n + 5)(4n + 7) 1 [2' 8(n + 1)(2n + 3) 1J_,1 (X))4dx (4n + 5) f  4(n + 1)(2n + 3)2 j k=1 1 X2 (x L)2 n 1 2 (1 x2)2(p())4 ( k2 d k=1k)l (4n + 5)(4n + 7)/(1 X )) 8(n + 1)(2n + 3) ,1 1 (2n + 2)2 (4n + 5) 1 4(n + 1)(2n + 3)2 (2n + 2)2 1 2z. 1 k=1 k 2n2 1 1 24 4n(n +1 ) 11  4 k2dZ 2n 2 (1 x2)(P(X))4d 4( + 1) X2)(p (xxk)2dx 2n+ (2n + k=l 2n2  (2n + 1) (4n + 5)(4n + 7) (1 x2)3p()4dx 8(n + 1)(2n + 3) _1 r[4n(n + 1) 2n2 (4n + 5) 21 (2n + 1)2 (2n + 1) 4(n + 1)(2n + 3)2 (2n + 2)2 (1 )(P,())4 ( k)2 1 k=1 k and from (5.3.3) we deduce 1 ) (X))4 3n3(4n + 5)(4n + 7) (1 2)3(pn())4d. 1(21 > 64(2n + 1)(n + 1)(2n + 3)  This proves Theorem 5.2. Now, as (1 x)(P'(x))4dx > (1 x2)2(P())4d > (1 x2)3(())4dx 1 1 1 we deduce (5.1.1) from (5.3.3). Similarly, we deduce (5.1.2) from (5.3.4) and (5.3.5). This proves the Corollary. D Proof of Theorem 5.3 Let P, E Hn and p > 2 even. Denote = J_' 11 Thus, From Identity 5.2.4 we see that pJr > ((p r )n (p r 2))Jr+2 (r = 0,2,...,p 4) which yields f1,(P'(x))Pdx f, (P,(x))Pdx Jo Jp Jo J2 Jp4 Jp2 \2 J4 Jp2 Jp > (p )n(p2) \ ~p / ((p 3)n (p 4) p 3n2 *\ p~ ((p 1)n (p 2))((p 3)n (p 4))...(3n 2) 2 f2 1((())) d P1 1 (P()) 2 P2 P f P ()) d2 pn(pn + 1) 4(p 2) 4(pn 2) pf+1 P 2 ((p 1)n (p 2))((p 3)n (p 4))...(3n 2)n(pn + 1) pt(pn 2) with the last inequality following from Theorem 1.4.4. Now, we use that St1'(1 t)ldt= r(p)F(q) 0 r(p + q)' where p and q are positive real numbers. To see that the result above is asymptotically sharp, we note that for the polynomial Po(x) = (1 x2)m (n = 2m) we have f ,(Po(x))Pdx f (Po(x))Pdx nP fl xP(12 dx fJ,(1 2)x7dx f), (P(X))p2 (P (Xn)2dx f (P.(x))Pdx nP fot 2 (1 t) 2 dt fo' t t)Ta St) F2 (P+3)dt nP y E(3) ( ... (2) (.1) F (222) ( ) 2 22 np(np2)...(np(2p2)) 2P p(np + 1)(np 1) (np (p 3))  n"(p 1)(p 3)...5 3 np(np 2) (np (2p 2)) This proves Theorem 5.3. CHAPTER 6 SUMMARY AND CONCLUSIONS 6.1 Synopsis The problems discussed in this dissertation come from the vast field of Approx imation Theory. The areas of this field considered here are Birkhoff Interpolation, Erdostype inequalities and Turantype inequalities. In Chapter Two, we proved the existence and uniqueness in the case of the 'modified' (0, 1,3,4) case of Birkhoff Interpolation on the zeros of 7r,i(x), and then provided an explicit representation in this case. Chapter Three presents the esti mates of the fundamental polynomials, which are then used to prove the uniform convergence of the 'modified' (0, 1,3,4) BirkhoffFej6r operator for the entire class of continuous functions on the interval [1,1]. This provides only the second known case of such a BirkhoffFejer operator. In Chapter Four, a classical theorem of P. Erdos [10] is extended in the L2 norm with the ultraspherical weight w(x) = (1 x2), a > 1, for the Lorentz class L, of polynomials. This result also generalizes some previously known partial results in the L2 norm. Then the result of Erd6s is extended in the L4 norm with weight w(x) = (1 x2)3. These results are the best possible, as the inequalities given are sharp. We note that this is the first sharp result extending the inequality of Erdos in the L4 norm. In Chapter Five, a classical theorem of P. Turn [45] is extended in the L2 norm with ultraspherical weight w(x) = (1 X2)1, a > 1 (a > 1 if Pn(l) = 0). Next, we extended the inequality of Turan in the L4 norm with weight w(x) = (1 x2)3. We note that this is the first sharp extension of the inequality of Turan in the L4 norm. These results provide lower bound analogues to the results given in Chapter Four for the class Hn of polynomials. These results are again the best possible, providing inequalities that are sharp. Finally, the classical theorems of P. Turan (in the uniform norm) and A.K. Varma [51] (in the L2 norm) are extended in the LP norm for p an even integer. This result is sharp in the asymptotic sense. 6.2 Open Problems We close this chapter by noting some open problems related to those presented in this work. First, we recall that the case of the 'modified' (0, 1,3,4) interpolation has been generalized into the 'modified' (0,... ,r 3, r 1,r) case. Similarly, the 'modified' (0,2), and the 'modified' (0,3) and (0,1,4) cases have been extended to the 'modified' (0,..., r 2, r) and (0,..., r 3, r) cases, respectively. In the paper of A. Sharma, J. Szabados, A.K. Varma, and the author [38], the problem of existence and uniqueness is settled, and explicit forms are provided. We note that the explicit representation of the last fundamental polynomials is found in the same manner as that given here. However, the remaining fundamental polynomials are determined in a different manner, one that is more amenable to the general situation. Nevertheless, better estimates are obtained from those given in this work. One expects that the (0, 3) and (0, 1,3, 4) cases are the only two of all the previously studied cases which converge uniformly for the entire class C[1, 1]. These two cases have a special 'balanced' nature. One problem would be to show that the given error of 0 w (f, log)) is sharp, and if not, to provide a sharper estimate. While it is easy to write down an explicit representation in the 'pure' (0, 1,3,4) case, having handled the 'modified' case, this representation is too complicated to be useful as a means of estimating the fundamen tal polynomials in the 'pure' case. Thus, convergence results cannot be determined. Hence, another problem would be to find an even simpler explicit representation in the 'pure' (or 'modified') (0, 1,3,4) case, and then provide convergence results in the 'pure' case. One would expect the error to be O(w4(f, I)). Another problem would be to find other BirkhoffFej6r operators which also converge uniformly for all continuous functions on the interval [1, 1]. Concerning Chapter Four, it would be interesting to provide the sharp inequal ity in the L4 norm when w(x) = 1, and in the more general case w(x) = (1 X2), a > 1, as well as providing sharp results in the LP norm for larger values of p. Also interesting is the extension of these results to higher derivatives. For the related extensions already known in the uniform norm, see W.A. Markov [25] (for all higher derivatives of algebraic polynomials of degree n) and J.T. Scheick [36] (for first and second derivatives of polynomials in the Lorentz class). In Chapter Five, again it would be interesting to provide the sharp inequality in the L4 norm when w(x) = 1, as well as when w(x) is the ultraspherical weight. Also, it would be interesting to provide the sharp inequality in the LP norm, p even, for each n, and also for higher derivatives. For the known results in the uniform norm for the class H,, see J. Er6d [12] (for the sharp result in the case of the first derivative) and V.F. Babenko and S.A. Pichugov [4] (for the case of the second derivative). REFERENCES [1] M. R. Akhlaghi, A. M. Chak and A. Sharma, (0,2,3) interpolation on the zeros of r,,(x), Approx. Theory and its Appl. 4 (1988), 5574. [2] M. R. Akhlaghi, A. M. Chak and A. Sharma, (0,3) interpolation on the zeros of 7,(x), Rocky Mountain J. Math. 19 (1989), 921. [3] K.E. Atkinson, An Introduction to Numerical Analysis, Wiley (1989). [4] V.F. Babenko and S.A. Pichugov, Inequalities for the derivatives of polynomials with real zeros, Ukrain. Math. J. 38 (1986), 347351. [5] J. Balzs and P. Turn, Notes on interpolation IIIV, Acta Math. Acad. Sci. Hungar. 8 (1957) 201205; 9 (1958), 195214; 243258. [6] S.N. Bernstein, Sur l'ordre de la meilleure approximation des functions continues par des polynomes de degree donnd, Memoires de l'Acad6mie Royale de Belgique 4(2) (1912), 1103. [7] S.N. Bernstein, Quelques remarques sur l'interpolation, Math. Ann. 79 (1918), 112. [8] G.D. Birkhoff, General meanvalue and remainder theorems with applications to mechanical differentiation and integration, Trans. Amer. Math. Soc. 1 (1906), 107136. [9] R.P. Boas, Jr., Inequalities for the derivatives of polynomials, Math. Mag. 42 (1969), 165174. [10] P. Erd6s, Extremal properties of polynomials, Ann. of Math. 4 (1940), 310313. [11] P. Erd6s and A. K. Varma, An extremum problem concerning algebraic polyno mials, Acta. Math. Hungar. 47 (1986), 137143. [12] J. ErSd, Bizonyos polinomok maximumdnak also korldtjdr6l, Math. Fiz. Lapok 46 (1939), 5882. [13] G. Faber, Ober die interpolatorische parstellung stetiger funktionen, Jber. Deutsch Math.Verein 23 (1914), 190210. [14] L Fej6r, Ueber interpolation, Nachr. Akad. Wiss. G6ttingen Math.Phys. K1. (1916), 6691. [15] G. Freud, Bemerkung iiber die Konvergenz eines Interpolationsverfahrens von P. Turdn, Acta Math. Acad. Sci. Hungar. 9 (1958), 337341. 82 [16] H. Gonska, Degree of approximation by lacunary interpolators: (0,..., R 2, R) interpolation, Rocky Mountain J. Math. 19 (1989), 157171. [17] C. H. Hermite, Sur la formula d'interpolation de Lagrange, J. Reine Angew Math. 84 (1878), 7079. [18] E. Hille, G. Szeg6 and J. D. Tamarkin, On Some Generalizations of a Theorem of A. A. Markov, Duke Math. Journal 3 (1937), 729739. [19] D. Jackson, The Theory of Approximation, Amer. Math Assoc., New York (1930). [20] N.D. Kazarinoff, Analytic Inequalities, Holt, Rinehart and Winston, New York (1961). [21] G.G. Lorentz, Bernstein Polynomials, Toronto (1953). [22] G.G. Lorentz, The degree of approximation by polynomials with positive coeffi cients, Math. Ann. 151 (1963), 239251. [23] G.G. Lorentz, K. Jetter and S. D. Riemenschneider, Birkhoff Interpolation, AddisonWesley, Reading, MA, 1983. [24] A.A. Markov, On a problem of D.I. Mendeleev (in Russian), Zap. Imp. Akad. Nauk 62 (1889), 124. [25] W.A. Markov, Ober Polynome, die in einem gegebenen Intervalle m6glichst wenig von Null abweichen, Math. Annalen 77 (1916), 218258. [26] D. Mendeleev, Investigation of aqueous solutions based on specific gravity (in Russian), St. Petersburg (1887). [27] T.M. Mills, Some techniques in Approximation Theory, Math. Scientist 5 (1980), 105120. [28] G. V. MilovanoviC and M. S. Petkovi6, Extremal Problems for Lorentz classes of nonnegative polynomials in L2 metric with Jacobi weight, Proc. Amer. Math. Soc. 102 (1988), 283289. [29] T.J. Rivlin, An Introduction to the Approximation of Functions, Dover Publi cations, Inc., New York, 1969. [30] C. Runge, Uber empirische Funktionen und die Interpolation zwischen aquidis tanten Ordinaten, Zeits fur Math. und Pkysik 46 (1901), 224243. [31] G. Sansone, Orthogonal Functions, R. E. Krieger Publ. Co., 1977. [32] R.B. Saxena, Convergence of interpolatory polynomials, (0,1,2,4) interpolation, Trans. Amer. Math. Soc. 95(2) (1960), 361385. [33] R.B. Saxena, Convergence in modified (0,2) interpolation, Magyar Tudo. Akad. 7(3) (1962), 255271. [34] R. B. Saxena and A. Sharma, On some interpolatory properties of Legendre polynomials, Acta Math. Acad. Sci. Hungar. 9 (1958), 345358. [35] R. B. Saxena and A. Sharma, Convergence of interpolatory polynomials, Acta Math. Acad. Sci. Hungar. 10 (1959), 157175. [36] J.T. Scheick, Inequalities for derivatives of polynomials of special type, J. Ap prox. Th. 6 (1972), 354358. [37] A. Sharma and D. Leeming, Lacunary interpolation (0, n 1, n) case, Math ematica 11(34) (1969), 155162. [38] A. Sharma, J. Szabados, B. Underhill and A. K. Varma, On some general lacunary interpolation problems (to appear in J. Approx. Th.). [39] J. Surinyi and P. Turn, Notes on interpolation I, Acta. Math. Acad. Sci. Hungaricae 6 (1955), 6779. [40] J. Szabados, On the order of magnitude of fundamental polynomials of Hermite interpolation, Acta Math. Hungar. 61 (1993), 357368. [41] J. Szabados, In memorial : Arun Kumar Varma (19341994), J. Approx. Th. 84 (1996), 111. [12] J. Szabados and A.K. Varma, On (0,1,2) interpolation in uniform metric, Proc. Amer. Math. Soc. 109 (1990), 975979. [43] J. Szabados and A. K. Varma, On convergent (0,3) interpolation processes, Rocky Mountain J. Math. 24 (1994), 729757. [44] G. Szego, Orthogonal Polynomials, AMS Coll. Publ., Vol. XXIII, Providence, RI, 1978. [45] P. Turn, Uber die ableitung von polynomen, Compositio Math. 7 (1939), 8995. [46] B. Taylor, The Method of Increments, (1715). [47] A. K. Varma, An analogue of some inequalities of P. Erdos and P. Turdn concerning algebraic polynomials satisfying certain conditions, Colloquia Math ematica Societatis Janos Bolyai 19, Fourier Analysis and Approximation Theory, Budapest, 1976. [48] A. K. Varma, An analogue of some inequalities of P. Turn concerning algebraic having all zeros inside [1, 1], Proc. Amer. Math. Soc. 55 (1976 no. 2), 305309. [49] A. K. Varma, An analogue of some inequalities of P. Turn concerning algebraic having all zeros inside [1, 1] II, Proc. Amer. Math. Soc. 69 (1978), 2533. [50] A. K. Varma, Some inequalities of algebraic polynomials having real zeros, Proc. Amer. Math. Soc. 75 (1979), 243250. [51] A. K. Varma, Some inequalities of algebraic polynomials having all zeros inside [1, 1], Proc. Amer. Math. Soc. 88 (1983 no. 2), 227233. [52] A. K. Varma, A. Saxena and R. B. Saxena, Lacunary interpolation: Modified (0, 1,4) case, Approximation Theory (Memphis, TN, 19991), 479501, Lecture Notes in Pure and Appl. Math. 138, Dekker, New York, 1992. [53] A. K. Varma and A. Sharma, Some interpolatory properties of Tchebicheff poly nomials, (0,2) case modified, Publ. Math. Debrecen 8 (1961), 336349. 84 [54] P. Vertesi, Notes on the convergence of (0,2) and (0, 1,3) interpolation, Acta Math. Acad. Sci. Hungar. 22 (19712), 127138. [55] K. Weierstrass, Uber die analytische Darstell barkeit sogennanter willkurlicher Funktionen reeler Argumente, Sitz. der Acad. Berlin (1885), 633639, 789805. [56] S. P. Zhou, On Turdn's inequality in the LP norm (in Chinese), J. Hangzhou Univ. 11 (1984), 2833. BIOGRAPHICAL SKETCH Brandon Underhill was born in Jacksonville, Florida, in August, 1969. He received his Bachelor of Arts degree with a minor in Classics from the University of Florida in August of 1990. He entered graduate school as a teaching assistant at the University of Florida that same month. He received his Master of Science degree from the University of Florida in August of 1992. He delivered an invited talk at the 1995 International Conference on Approximation Theory and Function Series in Budapest, Hungary, in August, 1995. I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. /Joseph lover Chairman ofessor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. S&/zK Li C. Shen Associate Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ermit N. Sigmon V Associate Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Stephen A. Saxon Professor of Mathematics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Pejaver V. Rao Professor of Statistics This dissertation was submitted to the Graduate Faculty of the Department of Mathematics in the College of Liberal Arts and Sciences and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. May 1996 Dean, Graduate School LD 1780 199 " UNIVERSITY OF FLORIDA 3 1262 08555 0605I 3 1262 08555 0605 
Full Text 
xml version 1.0 encoding UTF8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID E2JAAFLHL_1G3TPS INGEST_TIME 20131010T00:08:14Z PACKAGE AA00017658_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES 