Correspondence, pre-1960

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Correspondence, pre-1960
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Mathematics -- History -- 20th century

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University of Florida
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Prof. Dr. KONRAD KNOPP -(14b) TUBINGEN 27.4.1951.
Am Apfelberg 6


Dear Dr. Chung,
I thank you very much for your kind letter of April 21,1951
and especially for the proofs of your three theorems. They are
indeed note quite easy and are very interesting. Also many thanks
for the two new theorems. The first of them may be stated and
proved.in the following way:
Hyp. Z4,,= A is absolutely convergent, b 4 b, (all compl.)
n n
Concl. L a,4 6, -> A 4.
Proof. c. 4.. /, with 4~v 4. .-,
and the matrix (an) fulfills the three Toeplitz-conditiorLs

Z-.,J i .Z A 46 -o 4 4 Mvfxa4.
As you do not mention the receipt of my proof of the Abelian
theorem in question, I am afraid that my lel;ter may be host. I










I sent it (by ordinary mail) about 10 days later than my last air-
mail-letter, i.e. in the beginning or middle of February. At the
same time I sent a similar Manuscript, written in German, to Kara-
ml:ta for being published in hi Journal. Please let me know
whether you received meanwhile my ordinary mailletter with the
proof.

With kind regards
bcincerely yours






TUbingen, d. 14.Mai 1951


Dear Dr. Chung,

Thank you very much for your kind letter of April 30 -1951.
I send as printed matter a paper of mine (MZ 49, 1943, 219-255),
where you find details about L(t) = C(t) exp/ j ,r.
T
The theorem you mention now (If a > b > 0, b -9 b,
n= n= n
then 24.v Iv 4, -4v) Jis in this generality not correct: If
for instance a 4, (i o %, v o q~ 2 you have -90 but

and so "*
S- and not --) 0
The linear transformation

ft v with 4- s,., /C,...- 4.

is a Nbrlund transformation and is regular (permanent) if and
only if 40/(ae-*-.4.)1If the an are complex there is in addition
the condition l-,(~...* 4,.1 Mo (.4**#''** -.
In an early paper of mine (Rend. del circolo mat. di Palerm(
32, 1911, 95 110) I studied under various conditions sums of
the form C#"'..'#'4 (now called "Faltung" of the sequences
,(4. and {,4 ). Unfortunately I have not a single reprint
of this paper.
I am still puzzled with your theorem 3 of your MS you sent
me with your letter dated April 21 1951: In your first letter
(of January 3) you stated, that every diagonal series and every
column series is divergent. And now you prove the convergence.
But I do not understand how you get formula (4).

VWith kind regards
Scincerely yours



A/twx


Prof. Dr.. Konrad Knopp




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