# Differential operators and differential equations of by Pieter Cornelis. Sikkema

By Pieter Cornelis. Sikkema

Read or Download Differential operators and differential equations of infinite order with constant coefficients: Researches in connection with integral functions of finite order PDF

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Additional info for Differential operators and differential equations of infinite order with constant coefficients: Researches in connection with integral functions of finite order

Example text

These results needed we summarize in theorem 25 which, in fact, is a generalization of theorem 26. On account of the very general character of theorem 25 - as we know a certain class of "divergent" operators is also admitted, if the order of the given integral function h(x) is less than i - it is a far-going generalization of Whittaker's theorem I, in so far as in this theorem the integral function I (x) does not exceed the normal type of the order 1. 38 CHAPTER I NECESSARY AND SUFFICIENT CONDITIONS Necessary and sufficient conditions for a differential operator 0 F(D) _ a,iDn to be applicable to all integral functions that are of the same type of a fixed finite order will now be derived.

N < oo. n-. oo If the function y(x) is an integral function of the minimum type of 9) That the condition mentioned is not only sufficient but also necessary for a differential operator F(D) to be applicable to all integral functions y(x) that do not exceed the minimum type of the order a(a > 0), may be proved in an analogous way as in the similar case in theorem 1. See footnote 2). 7) For the notion "analytic at z = 0" see footnote 9) on page 21. 8) The proof of this assertion follows the proof of theorem 5.

In fact, if this should not be so, there would exist a positive integer Nk, such that for every it 1,'k the element d would satisfy 1 d k. n! Consequently 1 1 1 (n ! , {n ! 1 1 k) it = 1 for every is Z Nk. Hence formula (28) was not correct for S = --k which contradicts the fact, that (28) holds for every S > 0. -1. If k Z I the element dnk is in the sequence the element with the smallest subscript > nx_1, for which d,, > n! k+1. We now assert, that the sequence {dnk}, constructed in this way, possesses the three properties mentioned in the lemma.