The Journal of the Indian Mathematical Society

K. Chandrasekharan ¹, S. Minakshisundaram ²

Affiliations
1 Tata Institute of Fundamental Research, Bombay, IN
2 Andhra University, Waltair, IN

Abstract

This note has for its object the clarification of certain points in our book [1]. This clarification seems to us to be necessary in order to remove possible misconceptions regarding the validity of some of the results in [I] which are discussed in [2]; it will appear that those results are correct. We take this opportunity also to touch on some points other than those arising from [2], Chapter and page references, unless otherwise indioated, apply to the book.

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K. Chandrasekharan ¹

Affiliations
1 Madras University, IN

Abstract

The problem of summability of series of eigenfunctions of boundary-value problems has been solved by Dr. S. Minakshisundaram, by making use of the method of Bessel-summability developed by me in a paper which is not yet published. The object of this note is to answer the corresponding question of absolute summability.

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K. Chandrasekharan ¹

Affiliations
1 University of Madras, IN

Abstract

The method of Bessel summation of series which was developed by me in a previous paper is applied here to study the summability of the product of two given series. This product, however, is not the well-known Cauchy-product; I call it the Bessel-product.

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K. Chandrasekharan ¹

Affiliations
1 University of Madras, IN

Abstract

The Second Theorem of Consistency for summable series has been proved by Hardy in the following form: If

(i) the series Σc_n is summable (λ, k) to the sum C,

(ii) μ is a logarithmico-exponential function of λ such that μ = O(λ^ρ),

where Δ is a constant, then, the series Σc_n is summable (μ, k) to the sum C.

The object of this paper is to prove the corresponding theorem for absolutely summable series.

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K. Chandrasekharan ¹

Affiliations
1 University of Madras, IN

Abstract

In a paper published in 1927, Landau gave a new proof of Hadamard's theorem on the factorization of an integral function of finite order. His proof arose out of the study of the problem of factorization of (s-I)ζ(s), which had been for the first time solved by Hadamard and for which Ananda Rau gave an alternative proof by the use of Gauchy's theorem of residues. The object of this paper is to combine the methods of Landau and Ananda Rau to obtain another proof of Hadamard's general theorem of factorization.

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