A Tauberian theorem for power series

2001 ◽  
Vol 77 (4) ◽  
pp. 354-359 ◽  
Author(s):  
T. Hilberdink
Keyword(s):  
Power Series ◽  
Tauberian Theorem

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.

A Tauberian theorem for absolute summability

1970 ◽  
Vol 67 (2) ◽  
pp. 321-326 ◽  
Author(s):  
G. Das
Keyword(s):  
Power Series ◽  
Infinite Series ◽  
Tauberian Theorem ◽  
Power Series Expansion ◽  
Binomial Coefficient ◽  
Absolute Summability ◽  
Partial Sums ◽  
Function Of Bounded Variation ◽  
Image Position ◽  
The Given

Let be the given infinite series with {sn} as the sequence of partial sums and let be the binomial coefficient of zn in the power series expansion of the function (l-z)-σ-1 |z| < 1. Now let, for β > – 1,converge for 0 ≤ x < 1. If fβ(x) → s as x → 1–, then we say that ∑an is summable (Aβ) to s. If, further, f(x) is a function of bounded variation in (0, 1), then ∑an is summable |Aβ| or absolutely summable (Aβ). We write this symbolically as {sn} ∈ |Aβ|. This method was first introduced by Borwein in (l) where he proves that for α > β > -1, (Aα) ⊂ (Aβ). Note that for β = 0, (Aβ) is the same as Abel method (A). Borwein (2) also introduced the (C, α, β) method as follows: Let α + β ╪ −1, −2, … Then the (C, α, β) mean is defined by

A Tauberian theorem for power series

1954 ◽  
Vol 60 (1) ◽  
pp. 94-97
Author(s):  
William T. Reid
Keyword(s):  
Power Series ◽  
Tauberian Theorem

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.

scholarly journals A Tauberian theorem for a generalized power series method

2005 ◽  
Vol 18 (10) ◽  
pp. 1129-1133 ◽  
Author(s):  
Richard F. Patterson ◽  
Pali Sen ◽  
B.E. Rhoades
Keyword(s):  
Power Series ◽  
Tauberian Theorem ◽  
Series Method ◽  
Power Series Method ◽  
Generalized Power Series
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