Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Matrix transformations of power series

Authors: David Borwein and Amnon Jakimovski
Journal: Proc. Amer. Math. Soc. 122 (1994), 511-523
MSC: Primary 47B37; Secondary 30B10, 40C05
MathSciNet review: 1198451
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the sequence of transforms $ ({g_n})$ of a power series $ \sum\nolimits_{n = 0}^\infty {{a_n}{z^n}} $ given by $ {g_n}(z): = \sum\nolimits_{k = 0}^\infty {{b_{nk}}{a_k}{z^k}} $. We establish necessary and sufficient conditions on the matrix $ ({b_{nk}})$ for the sequence $ ({g_n})$ to converge uniformly on compact subsets of the disk $ {D_P}: = \{ z:\vert z\vert < P\} $ to a function holomorphic on $ {D_P}$.

References [Enhancements On Off] (What's this?)

  • [1] E. Hille, Analytic function theory, Blaisdell, New York, 1963.
  • [2] M. F. Lejá, Sur la sommation des séries entières par la méthode des moyennes, Bull. Sci. Math. (2) 54 (1930), 239-245.
  • [3] W. Luh, Über die Nörlund-Summierbarkeit von Potenzreihen, Period. Math. Hungar. 5 (1974), 47–60 (German). MR 0367507,
  • [4] Wolfgang Luh, Summierbarkeit von Pontenzreihen—notwendige Bedingungen, Mitt. Math. Sem. Giessen 111 (1974), 48–67 (German). MR 0364937
  • [5] A. Peyerimhoff, Lectures on summability, Lecture Notes in Mathematics, Vol. 107, Springer-Verlag, Berlin-New York, 1969. MR 0463744
  • [6] K. Stadtmüller, Summability of power series by non-regular Nörlund methods, J. Approx. Theory 68 (1991), 33-44.
  • [7] E. C. Titchmarsh, The theory of functions, Oxford Univ. Press, London, 1947.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47B37, 30B10, 40C05

Retrieve articles in all journals with MSC: 47B37, 30B10, 40C05

Additional Information

Keywords: Nörlund, matrix transforms, power series
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society