Expansions of Arbitrary Analytic Functions in Series of Exponentials

Dickson, D. G.

Canadian journal of mathematics, Tome 33 (1981) no. 2, pp. 347-356

Résumé

Let φ ≠ 0 be an entire function of one complex variable and of exponential type. Let B denote the set of all monomial exponentials of the form zneζ where ζ is a zero of φ of order greater than h. If R is a simply connected plane region and H(R) denotes the space of functions analytic in R with the topology of uniform convergence on compacta, then φ can be considered as an element of the topological dual H′(R) if the Borel transform of φ is analytic on , the complement of R. The duality is given by where C is a simple closed curve in the common region of analyticity of ƒ and , and C winds once around the complement of a set in which is analytic.

Détail

DOI : 10.4153/CJM-1981-028-0

Bibliographie

[1] 1. Dickson, D. G., Expansions in series of solutions of linear difference-differential and infinité order differential equations with constant coefficients, Mem. Amer. Math. Soc. 23 (1957), 860. Google Scholar

[2] 2. Dickson, D. G., Analytic mean periodic functions, Trans. Amer. Math. Soc. 110 (1964), 361–374. Google Scholar

[3] 3. Dzjadyk, V. K., On convergence conditions for Dirichlet series on closed polygons, Mat. Sb. 95(173) (1974), 475–493. Google Scholar

[4] 4. Dzjadyk, V. K. and Krutigolova, E. K., The representation of analytic functions by a Dirichlet series at the boundary of the convergence domain, Mat. Zametki 14 (1973), 796–780. Google Scholar

[5] 5. Krutigolova, E. K., The behavior of a Dirichlet series on the boundary of a region of convergence, Ukrain. Mat. Z. 27 (1975), 234–240. Google Scholar

[6] 6. Krutigolova, E. K., Representation of analytic functions by Dirichlet series on the boundaries of closed convex polygon domains, Ukrain. Mat. Z. 27 (1975), 516–521. Google Scholar

[7] 7. Leont'ev, A. F., Representation of arbitrary functions by Dirichlet series, Dokl. Nauk SSSR 164 (1965), 40–42. Google Scholar

[8] 8. Leont'ev, A. F., On the representation of functions by sequences of Dirichlet polynomials, Mat. Sb. 70 (112) (1966), 132–144. Google Scholar

[9] 9. Leont'ev, A. F., Qn jfoe representation of analytic functions by Dirichlet series, Mat. Sb. 80 (122) (1969), 117–156. Google Scholar

[10] 10. Leont'ev, A. F., Representation of functions by generalized Dirichlet series, Uspehi Mat. Nauk 4 (1969), 97–164. Google Scholar

[11] 11. Leont'ev, A. F., On conditions of expandibility of analytic functions in Dirichlet series, Izv. Akad. Nauk SSSR 36 (1972), 1282–1295. Google Scholar

[12] 12. Leont'ev, A. F., On the representation of analytic functions in a closed convex region by a Dirichlet series, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 577–592. Google Scholar

[13] 13. Leont'ev, A. F., The representation of analytic functions in a polygonal convex closed domain by Dirichlet series, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 127–137. Google Scholar

[14] 14. Moore, M. G., On expansions in series of exponential functions, Amer. J. Math. 62 (1940), 83–90. Google Scholar

[15] 15. Sedletskii, A. M., Expansions of functions into Dirichlet series on closed convex polygons, Siberian Math. J. 19 (1979), 622–629. Google Scholar