On certain families of generalized Nörlund methods and power series methods
type: Item
instance of: scholarly article
title: On certain families of generalized Nörlund methods and power series methods
zbMATH Open document ID: 0934.40003
author: Anne Tali
author: Ulrich Stadtmüller
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-04-25T00:00:00Z
review text: A generalized Nörlund method is generated by a sequence \(\{r_n\}\), where \(r_n=(a*b)_n:=\sum^n_{k=0}a_{n-k}b_k\), and is denoted by \((N,a, b)\). A sequence \(\{s_n]\) is summable to \(s\) by a power series summability method \(J_q\), if the power series \(q(x)=\sum^\infty_{n=0}q_nx^n\) has a radius of convergence of \(1\), \(q_s(x): =\sum^\infty_{n=0} s_nq_nx^n\) converges for \(|x|<1\) and \((1/q(x)) \sum^\infty_{n=0} s_nq_nx^n\to s\) as \(x\to 1-\). The extended power series method \(J_q^*\) has the property that \(q_s(x)\) has a positive radius of convergence, can be holomorphically extended along \((0,1)\) and \(q_s(x)/q(x)\to s\) as \(x\to 1-\). The authors obtain inclusion relations for \(J_q\), \(J_q^*\), \(J_r\cdot (N,a,q)\), and \((N,a,q)\cdot J_r\). A family of summability methods \((A_\alpha\}\) is called a Cesàro-type family if the methods \(A_\alpha\) and \(A_\alpha'\), for any \(\alpha_0<\alpha <\alpha'\), are related by the equation \(\eta_n^{\alpha'}=(1/b_n^{\alpha'})\sum^n_{k=0}A_{n-k}^{\alpha'- \alpha-1} b_k^\alpha \eta_k^\alpha\). The authors obtain inclusion and Tauberian theorems for certain matrices \(A_\alpha\). They also obtain inclusion and Tauberian theorems for generalized Nörlund matrices of the form \((N,p^{\alpha \beta},q)\), where \(p_n^{\alpha \beta}:=(p^{\alpha,\beta-1}*p)_n\).
Mathematics Subject Classification ID: 40G05
Mathematics Subject Classification ID: 40D25
Mathematics Subject Classification ID: 40E05
zbMATH DE Number: 1354749
zbMATH Keywords: generalized Nörlund method
zbMATH Keywords: power series summability method
zbMATH Keywords: radius of convergence
zbMATH Keywords: inclusion and Tauberian theorems
reviewed by: Billy E. Rhoades
MaRDI profile type: Publication
full work available at URL: jmaa.1999.6503
OpenAlex ID: W2058021539
cites work: Q4726487
cites work: Q3251561
cites work: On Products of Sequences
cites work: Convexity theorems for Nörlund and strong Nörlund summability
cites work: ON SOME METHODS OF SUMMABILITY (II)
cites work: On two new methods of summability
cites work: Q5788558
cites work: General Nörlund transforms and power series methods
cites work: On Scales of Summability Methods
cites work: The law of the iterated logarithm for certain power series and generalized Nörlund methods
cites work: Tauberian- and Convexity Theorems for Certain (N,p,q)-Means
cites work: Q3215718
cites work: Q2996562
cites work: Q4951451
cites work: Tauber-Sätze für Potenzreihenverfahren. (Tauberian theorems for power series methods)
DOI: 10.1006/JMAA.1999.6503
description: scientific article; zbMATH DE number 1354749