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  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 86, Issue 1-2
  • January 1980, pp. 85-101

Differential Operators of infinite order

  • Einar Hille (a1)
  • DOI: http://dx.doi.org/10.1017/S0308210500012026
  • Published online: 14 November 2011
Abstract
Synopsis

The differential operators in question are of the form G(DZ) where G(w)is an entire function of order at most 1/n and minimal type while Dz is a linear differential operator of order n with coefficients which are entire ( = integral) functions of z, usually polynomials. This class of operators form a natural generalization of the class G(d/dz) studied during the first half of the century Muggli, Polya, Ritt and others. The class G(DZ) was introduced by the present author and his pupils in the 1940s. In fact, the present paper is partly based on a MS from that period, mostly devoted to the special case

but also containing generalizations, some of which were later worked out by Klimczak. A basic tool in this paper is the characteristic series

Examples are given showing that the domain of absolute convergence of such a series need neither be convex nor of finite connectivity, a question which has puzzled the author for forty odd years. Characteristic series arising from regular or singular boundary value problems for the operator Dz are used to study the inversion problem

for given F(z). In particular it is shown that exp (Dx)[W(z)] = 0 has the unique solution W(z) ≡ 0. Some singular boundary value problems are considered briefly.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

2Einar Hille . Note on Dirichlet's series with complex exponents. Ann. Math. 25 (1924), 261275.

2AEinar Hille . Contributions to the theory of Hermitian series. Duke Math. J. 5 (1939), 895936.

3Einar Hille . A class of differential operators of infinite order. Duke Math. J. 7 (1940), 458495.

7Einar Hille . Ordinary Differential Equations in the Complex Plane (New York: Wiley, 1976).

8Einar Hille . On a special Weyl-Titchmarsh problem. Integral Equations and Operator Theory 1 (1978), 215225.

9Walter J. Klimczak . Differential operators of infinite order. Duke Math. J. 20 (1953), 295319.

11Hermann Muggii . Differentialgleichungen unendlich höher Ordnung mit konstanten Koeffizienten. Comment. Math. Heb. 11 (1938), 151159.

14Georg Polya . Untersuchungen über Lücken und Singularitäten von Potenzreihen. Math. Z. 29 (1929), 549640.

15J. F. Ritt . On a general class of linear homogeneous differenital equations of infinite order with constant coefficients. Trans. Amer. Math. Soc. 18 (1917), 2749.

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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