XXII.—On Methods of Summability Based on Power Series

1957 ◽  
Vol 64 (4) ◽  
pp. 342-349
Author(s):  
D. Borwein
Keyword(s):  
Power Series ◽  
Summability Method ◽  
Radius Of Convergence ◽  
Main Concern ◽  
Image Position

SynopsisGiven a power series with real non-negative coefficients and having radius of convergence p, a summability method P is defined as follows:The main concern of this note is to establish conditions sufficient for one such method to include another.

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

A generalization of the Hurwitz composition theorem to irregular power series

1951 ◽  
Vol 47 (3) ◽  
pp. 477-482 ◽  
Author(s):  
H. G. Eggleston
Keyword(s):  
Power Series ◽  
Radius Of Convergence ◽  
Finite Radius ◽  
Image Position ◽  
Composition Theorem

When two functions are given, each with a finite radius of convergence, a theorem due independently to Hurwitz and Pincherle (1, 2) provides information about the position of the singularities of the functionin terms of the positions of the singularities of f(z) and g(z).

scholarly journals Arithmetic properties of lacunary power series with integral coefficients

1965 ◽  
Vol 5 (1) ◽  
pp. 56-64 ◽  
Author(s):  
K. Mahler
Keyword(s):  
Power Series ◽  
Radius Of Convergence ◽  
Arithmetic Properties ◽  
Image Position ◽  
Infinite Sequences

This note is concerned with arithmetic properties of power series with integral coefficients that are lacunary in the following sense. There are two infinite sequences of integers {rn} and {sn}, satisfying such that It is also assumed that f(z) has a positive radius of convergence, Rf say, where naturally . A power series with these properties will be called admissible.

Notes on the theory of series (XXIV): a curious power-series

1946 ◽  
Vol 42 (2) ◽  
pp. 85-90 ◽  
Author(s):  
G. H. Hardy ◽  
J. E. Littlewood
Keyword(s):  
Power Series ◽  
Radius Of Convergence ◽  
Image Position

1. This note originates from a question put to us by Mr W. R. Dean concerning series of the typewhere θ is irrational. Series of the typeare familiar: it is well known, for example, thatmay have any radius of convergence from 0 to 1 inclusive, according to the arithmetic nature of θ. It is natural to ask how this radius is connected with that ofand, more generally, how those of (1·1) and (1·2) are connected.

VI.—On Solvability of Some Two-Parameter Eigenvalue Problems in Hilbert Space

1968 ◽  
Vol 68 (1) ◽  
pp. 83-93
Author(s):  
Z. Bohte
Keyword(s):  
Hilbert Space ◽  
Power Series ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Radius Of Convergence ◽  
Finite Radius ◽  
Image Position ◽  
Adjoint Operators ◽  
Two Parameter

SynopsisThis paper studies two particular cases of the general 2-parameter eigenvalue problem namelywhere A, B, B1, B2, C, C1, C2 are self-adjoint operators in Hilbert space, all except A being bounded. The disposable parameters λ and μ have to be determined so that the equations have non-trivial solutions x, y.On the assumption that the solution is known for ∊ = o, solutions are constructed in the form of series for λ, μ, x, y as power series in ∊ with finite radius of convergence.

Note on a Divergent Series

1941 ◽  
Vol 37 (1) ◽  
pp. 1-8 ◽  
Author(s):  
G. H. Hardy
Keyword(s):  
Power Series ◽  
Asymptotic Series ◽  
Radius Of Convergence ◽  
Divergent Series ◽  
Image Position

1. The seriesis the simplest and most familiar power series whose radius of convergence is zero. It is natural to regard it as a development of the function G(z) defined, whenbyForsay;andoraccording as x is positive or negative. Thus the series (1·1) is an asymptotic series for G(z) in the sense of Poincaré.

Entire Functions with Some Derivatives Univalent

1974 ◽  
Vol 26 (1) ◽  
pp. 207-213 ◽  
Author(s):  
S. M. Shah ◽  
S. Y. Trimble
Keyword(s):  
Power Series ◽  
Entire Functions ◽  
Radius Of Convergence ◽  
Open Disc ◽  
Image Position ◽  
Positive Integers ◽  
The Relationship ◽  
Derivatives Of

This paper is a continuation of the author's previous work, [6; 7], on the relationship between the radius of convergence of a power series and the number of derivatives of the power series which are univalent in a given disc.In particular, let D be the open disc centered at 0, and let f be regular there. Suppose that is a strictly-increasing sequence of positive integers such that each f(np) is univalent in D. Let R be the radius of convergence of the power series, centered at 0, that represents f. In [7], we investigated the connection between R and . We showed that, in general

On the Relation Between [f, g] and Aλ Summability

1974 ◽  
Vol 26 (4) ◽  
pp. 783-793
Author(s):  
Joaquin Bustoz
Keyword(s):  
Power Series ◽  
General Class ◽  
Inverse Function ◽  
Radius Of Convergence ◽  
Summability Methods ◽  
Image Position

First we will briefly define the [f, g] and Aλ summability methods. Let K = {w : |w| < 1}. T. H. Gronwall [3] introduced a general class of summability methods each of which involves a pair of functions f and g with the following properties. The function z = f (w) is analytic on \{1}, continuous and univalent on , with f (0) = 0, f (1) = 1, |f(w)| < 1 if w ∊ K. The inverse function w = f-1(z) is analytic on f (K)\{1}, and at z = 11.1where γ ≧ 1, a > 0, and the quantity in brackets is a power series in 1 — z with positive radius of convergence.

scholarly journals Approximation by a power series summability method of Kantorovich type Szász operators including Sheffer polynomials

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Valdete Loku ◽  
Naim L. Braha ◽  
Toufik Mansour ◽  
M. Mursaleen
Keyword(s):  
Power Series ◽  
Summability Method ◽  
Approximation Properties ◽  
Type Result ◽  
Szász Operators ◽  
Sheffer Polynomials

AbstractThe main purpose of this paper is to use a power series summability method to study some approximation properties of Kantorovich type Szász–Mirakyan operators including Sheffer polynomials. We also establish Voronovskaya type result.

Analogue of a theorem of Khintchine in fields of formal power series

1966 ◽  
Vol 62 (4) ◽  
pp. 637-642 ◽  
Author(s):  
T. W. Cusick
Keyword(s):  
Power Series ◽  
Positive Constant ◽  
Classical Result ◽  
Real Numbers ◽  
Linear Forms ◽  
Absolute Value ◽  
The Difference ◽  
Image Position ◽  
Formal Power ◽  
Positive Integers

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and letbe any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such thatfor all integers x1, x2, …, xm not all zero.

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