On the Partial Sums of Harmonic Developments and of Power Series

1942 ◽  
Vol 52 (1) ◽  
pp. 12
Author(s):  
Otto Szasz
Keyword(s):  
Power Series ◽  
Partial Sums

A Multiplicative Analogue of Schur's Tauberian Theorem

2003 ◽  
Vol 46 (3) ◽  
pp. 473-480 ◽  
Author(s):  
Karen Yeats
Keyword(s):  
Power Series ◽  
Asymptotic Behaviour ◽  
Dirichlet Series ◽  
Tauberian Theorem ◽  
Partial Sums ◽  
Regularly Varying Functions ◽  
Regularly Varying

AbstractA theorem concerning the asymptotic behaviour of partial sums of the coefficients of products of Dirichlet series is proved using properties of regularly varying functions. This theorem is a multiplicative analogue of Schur's Tauberian theorem for power series.

scholarly journals On (J, pn)-summability of Fourier series

1972 ◽  
Vol 18 (1) ◽  
pp. 13-17
Author(s):  
F. M. Khan
Keyword(s):  
Fourier Series ◽  
Power Series ◽  
Radius Of Convergence ◽  
Partial Sums ◽  
Summability Of Fourier Series ◽  
Image Position

Let pn>0 be such that pn diverges, and the radius of convergence of the power seriesis 1. Given any series σan with partial sums sn, we shall use the notationand

The zeros of the partial sums of power series

1963 ◽  
Vol 30 (4) ◽  
pp. 533-540 ◽  
Author(s):  
T. Ganelius
Keyword(s):  
Power Series ◽  
Partial Sums

scholarly journals On partial sums of Lagrange's series with application to the theory of queues

1963 ◽  
Vol 3 (4) ◽  
pp. 488-490 ◽  
Author(s):  
P. D. Finch
Keyword(s):  
Power Series ◽  
Partial Sums ◽  
Lagrange's Series ◽  
Lagrange's Theorem

Lagrange's theorem on the reversion of power series may be stated in the following form (e.g. Whittker and Watson [3]).

RESTRICTED UNIVERSALITY OF POWER SERIES

2001 ◽  
Vol 33 (5) ◽  
pp. 543-552 ◽  
Author(s):  
JEAN-PIERRE KAHANE ◽  
ANTONIOS D. MELAS
Keyword(s):  
Power Series ◽  
Finite Union ◽  
Radius Of Convergence ◽  
Nonempty Interior ◽  
Partial Sums ◽  
Universal Approximation ◽  
Compact Set ◽  
Approximation Properties

We prove the existence of a power series having radius of convergence 0, whose partial sums have universal approximation properties on any compact set with connected complement that is contained in a finite union of circles centred at 0 and having rational radii, but do not have such properties on any compact set with nonempty interior. This relates to a theorem of A. I. Seleznev.

Universal Power Series in ℂN

2011 ◽  
Vol 54 (2) ◽  
pp. 230-236 ◽  
Author(s):  
Raphaël Clouâtre
Keyword(s):  
Power Series ◽  
Holomorphic Function ◽  
Compact Subset ◽  
Partial Sums

AbstractWe establish the existence of power series in ℂN with the property that the subsequences of the sequence of partial sums uniformly approach any holomorphic function on any well chosen compact subset outside the set of convergence of the series. We also show that, in a certain sense, most series enjoy this property.

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