Efficient Analytic Continuation of Power Series by Matrix Summation Methods

2003 ◽  
Vol 2 (1) ◽  
pp. 137-153 ◽  
Author(s):  
Norair Arakelian ◽  
Wolfgang Luh
Keyword(s):  
Power Series ◽  
Analytic Continuation ◽  
Summation Methods ◽  
Matrix Summation

scholarly journals Polynomially bounded multisequences and analytic continuation

1982 ◽  
Vol 23 (1) ◽  
pp. 41-52
Author(s):  
Daniel J. Troy
Keyword(s):  
Power Series ◽  
Analytic Continuation ◽  
Linear Functional ◽  
Necessary And Sufficient Condition ◽  
Convergence Criteria ◽  
Dimensional Torus ◽  
Unit Factor ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Class Of Functions

Given a polynomially bounded multisequence {fm}, where m = (m1, …, mk) ∈ ℤk, we will consider 2k power series in exp(iz1), …, exp(izk), each representing a holomorphic function within its domain of convergence. We will consider this same multisequence as a linear functional on a class of functions defined on the k-dimensional torus by a Fourier series, , with the proper convergence criteria. We shall discuss the relationships that exist between the linear functional properties of the multisequence and the analytic continuation of the holomorphic functions. With this approach we show that a necessary and sufficient condition that the multisequence be given by a polynomial is that each of the power series represents, up to a unit factor, the same function that is entire in the variables

On exact constants for matrix summation methods

1995 ◽  
Vol 36 (4) ◽  
pp. 800-806
Author(s):  
L. P. Falaleev
Keyword(s):  
Summation Methods ◽  
Matrix Summation ◽  
Exact Constants

scholarly journals The a-points of Faber polynomials for a special function

1970 ◽  
Vol 11 (1) ◽  
pp. 1-6
Author(s):  
Hassoon S. Al-Amiri
Keyword(s):  
Analytic Function ◽  
Power Series ◽  
Analytic Continuation ◽  
Special Function ◽  
Laurent Expansion ◽  
Faber Polynomials ◽  
Analytic Element ◽  
Formal Expansion ◽  
Image Position

Let f(ζ) be a power series of the formwhere lim sup |an|1/n < ∞. The Faber polynomials {fn(ζ)} (n = 0, 1, 2, …) are the polynomial parts of the formal expansion of (f(ζ))n about ζ = ∞. Series (1) defines an analytic element of an analytic function which we designate as w = f(ζ). Since at ζ = ∞ the analytic element is univalent in some neighborhood of infinity; thus the inverse of this element is uniquely determined in some neighborhood of w= ∞, and has a Laurent expansion of the formwhere lim sup |bn|1/n = p < ∞. Let ζ = g(w) be this single-valued function defined by (2) in |w| > p. No analytic continuation of g(w) will be considered.

Hausdorff dimension of the set of almost convergent sequences

2022 ◽  
pp. 1-7
Author(s):  
Alexandr Usachev
Keyword(s):  
Hausdorff Dimension ◽  
Summation Method ◽  
Binary Representation ◽  
Hausdorff Dimensions ◽  
Summation Methods ◽  
Wide Range ◽  
Matrix Summation ◽  
Convergent Sequences

Abstract The paper deals with the sets of numbers from [0,1] such that their binary representation is almost convergent. The aim of the study is to compute the Hausdorff dimensions of such sets. Previously, the results of this type were proved for a single summation method (e.g. Cesàro, Abel, Toeplitz). This study extends the results to a wide range of matrix summation methods.

On the analytic continuation of functions defined by regrouped power series

1974 ◽  
Vol 15 (5) ◽  
pp. 408-412
Author(s):  
V. A. Belyaev
Keyword(s):  
Power Series ◽  
Analytic Continuation
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