Cesàro Means of Power Series (2)

1936 ◽  
Vol s2-40 (1) ◽  
pp. 345-352 ◽  
Author(s):  
A. E. Gwilliam
Keyword(s):  
Power Series ◽  
Cesàro Means ◽  
Cesaro Means ◽  
Cesáro Means

Uniform summability of power series

1972 ◽  
Vol 71 (2) ◽  
pp. 335-341 ◽  
Author(s):  
J. C. Kurtz ◽  
W. T. Sledd
Keyword(s):  
Banach Space ◽  
Power Series ◽  
Normal Matrix ◽  
Cesàro Means ◽  
Summability Factors ◽  
Cesaro Means ◽  
Space Topology ◽  
Nörlund Means ◽  
Summability Field ◽  
Cesáro Means

AbstractIt is shown that for the Cesàro means (C, α) with α > - 1, and for a certain class of more general Nörlund means, summability of the series σan implies uniform summability of the series σan zn in a Stolz angle at z = 1.If B is a normal matrix and (B) denotes the series summability field with the usual Banach space topology, then the vectors {ek} (ek = {0,0,..., 1,0,...}) are said to form a Toplitz basis for (B) relative to a method H if H — Σakek = a for each a = {ak}ε(B). It is shown for example that the above relation holds for B = (C,α), α> − 1 , and H = Abel method; also for B = (C,α) and H = (C,β) with 0 ≤ α ≤ β.Applications are made to theorems on summability factors.

Tauberian- and Convexity Theorems for Certain (N,p,q)-Means

1994 ◽  
Vol 46 (5) ◽  
pp. 982-994 ◽  
Author(s):  
Rüdiger Kiesel ◽  
Ulrich Stadtmüller
Keyword(s):  
Power Series ◽  
Large Class ◽  
Cesàro Means ◽  
Convexity Theorem ◽  
Series Method ◽  
Power Series Method ◽  
Cesaro Means ◽  
Nörlund Means ◽  
Cesáro Means ◽  
Power Series Methods

AbstractThe summability fields of generalized Nörlund means (N,p*α,p), α ∈ Ν, are increasing with a and are contained in that of the corresponding power series method (P,p). Particular cases are the Cesàro- and Euler-means with corresponding power series methods of Abel and Borel. In this paper we generalize a convexity theorem, which is well-known for the Cesàro means and which was recently shown for the Euler means to a large class of generalized Nörlund means.

Theorems Concerning Cesàro Means of Power Series

1934 ◽  
Vol s2-36 (1) ◽  
pp. 516-531 ◽  
Author(s):  
G. H. Hardy ◽  
J. E. Littlewood
Keyword(s):  
Power Series ◽  
Cesàro Means ◽  
Cesaro Means ◽  
Cesáro Means

scholarly journals Lebesgue points of $$\ell _1$$-Cesàro summability of d-dimensional Fourier series

2021 ◽  
Vol 6 (3) ◽  
Author(s):  
Ferenc Weisz
Keyword(s):  
Fourier Series ◽  
Lebesgue Point ◽  
Cesàro Means ◽  
Cesàro Summability ◽  
Cesaro Means ◽  
Cesaro Summability ◽  
Lebesgue Points ◽  
Cesáro Means

AbstractWe generalize the classical Lebesgue’s theorem and prove that the $$\ell _1$$ ℓ 1 -Cesàro means of the Fourier series of the multi-dimensional function $$f\in L_1({{\mathbb {T}}}^d)$$ f ∈ L 1 ( T d ) converge to f at each strong $$\omega $$ ω -Lebesgue point.

Growth conditions on Cesàro means of higher order

2013 ◽  
Vol 79 (3-4) ◽  
pp. 545-581
Author(s):  
Laurian Suciu ◽  
Jaroslav Zemánek
Keyword(s):  
Growth Conditions ◽  
Higher Order ◽  
Cesàro Means ◽  
Cesaro Means ◽  
Cesáro Means
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