On efficient analytic continuation of power series

N. U. Arakelian

Sbornik. Mathematics, Tome 52 (1985) no. 1, pp. 21-39

Résumé

The problem of efficiency of analytic continuation, which arose as a result of critical analysis of difficulties of Weierstrass' approach to the foundations of the theory of analytic functions, was a subject of numerous classical studies (Hadamard, Borel, LeRoy, Mittag-Leffler, Lindelöf, Pólya, et al.). The author discusses two questions related to the problem. First, by employing results of the theory of approximation by entire functions, the author succeeds in getting, roughly speaking, a converse to the well-known theorem of LeRoy and Lindelöf on analytic continuation of power series into angular domains, with further generalization and refinement of it. Second, the author discusses the question of efficiency of summation methods as applied to power series outside their circle of convergence. It is proved that the classical Mittag-Leffler–Lindelöf conditions of generalized star-likeness on the domain of summability are actually necessary. Bibliography: 19 titles.