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.

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.

scholarly journals Degree of approximation of a function by Nörlund means of its Fourier series

1981 ◽  
Vol 23 (3) ◽  
pp. 395-412
Author(s):  
R. B. Saxena
Keyword(s):  
Fourier Series ◽  
Degree Of Approximation ◽  
Cesàro Means ◽  
Cesaro Means ◽  
Nörlund Means ◽  
Cesáro Means ◽  
Approximation Of A Function

Two theorems of T.M. Flett [Quart. J. Math. Oxford Ser. (2) 7 (1956), 81–95] on the degree of approximation to a function by the Cesàro means of its Fourier series are extended to Nörlund means. Their conjugate analogues are also proved.

An O-Tauberian theorem and a high indices theorem for power series methods of summability

1994 ◽  
Vol 115 (2) ◽  
pp. 365-375 ◽  
Author(s):  
David Borwein ◽  
Werner Kratz
Keyword(s):  
Power Series ◽  
Tauberian Theorem ◽  
Series Method ◽  
Power Series Method ◽  
Power Series Methods

AbstractWe improve known Tauberian results concerning the power series method of summability Jp based on the sequence {pn} by removing the condition that pn be asymptotically logarithmico-exponential. We also prove an entirely new Tauberian result for rapidly decreasing pn.

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 Conditions for the equivalence of power series and discrete power series methods of summability

Filomat ◽  
2015 ◽  
Vol 29 (10) ◽  
pp. 2275-2280
Author(s):  
Sefa Sezer ◽  
İbrahim Çanak
Keyword(s):  
Power Series ◽  
Series Method ◽  
Power Series Method ◽  
Regularity Results ◽  
Power Series Methods

Discrete power series methods were introduced and their regularity results were developed by Watson [Analysis (Munich), 18(1): 97-102, 1998]. It was shown by Watson that discrete power series method (P?) strictly includes corresponding power series method (P). In the present work we present theorems showing when (P?) and (P) are equivalent methods and when two discrete power series methods are equivalent.

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.

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