On the kernel of sequence transformations
type: Item
instance of: scholarly article
title: On the kernel of sequence transformations
zbMATH Open document ID: 0819.65004
DOI: 10.1016/0168-9274(94)00054-9
published in: Applied Numerical Mathematics
publication date: 1995-09-05T00:00:00Z
review text: The paper deals with the production of constant sequences by use of a transformation based upon difference sequences. Let \({\mathcal G}\) be a system of sequences \({\mathcal G} [i] (i \geq 0)\), the members of \({\mathcal G} [i]\) being \(G(i | n)\) \((n \geq 0)\), \({\mathcal P}\) and \({\mathcal Q}\) be two sequences of integers \(p(n) \geq 0\) \((n \geq 0)\) and \(q(n)\) \((n \geq 0)\) and \({\mathcal F}\) be the sequence whose terms are \(F(n)\) \((n \geq 0)\). The members of the difference sequence \(\delta ({\mathcal F}, {\mathcal G} : {\mathcal P}, {\mathcal Q})\) of \({\mathcal F}\) are \(\delta ({\mathcal F}, {\mathcal G} : {\mathcal P}, {\mathcal Q} | n) = \sum F(i + n) G(i | n)\) \((- p(n) \leq i \leq q(n))\), being defined for \(n \geq 0\), where \(F(i + n) = 0\) when \(i + n < 0\). \(({\mathcal G}, {\mathcal P}\) and \({\mathcal Q}\) are fixed throughout: \(\delta ({\mathcal F}, {\mathcal G} : {\mathcal P}, {\mathcal Q})\) is written as \(\delta ({\mathcal F}).)\) \(L\) being constant, a relationship of the form \(\delta (L{\mathcal F}) = L \delta ({\mathcal F})\) evidently holds. \({\mathcal S}\) and \({\mathcal E}\) being sequences whose members are \(S(n)\) and \(E(n)\) respectively \((n \geq 0)\), the sequence \({\mathcal S} {\mathcal E}\) is defined by its members \(S(n) E(n)\) \((n \geq 0)\). Assuming no member of \(\delta ({\mathcal E})\) to be zero, the members of \({\mathcal T} ({\mathcal S} : {\mathcal E})\), the transform of \({\mathcal S}\) by \({\mathcal E}\), are \({\mathcal T} ({\mathcal S} : {\mathcal E} | n) = \delta ({\mathcal S} {\mathcal E} | n)/ \delta ({\mathcal E} | n)\) \((n \geq 0)\). \({\mathcal A}\) being the sequence whose members are \(\{L - S(n)\} E(n)\) \((n \geq 0)\), it is shown that if \(\delta ({\mathcal A} | n) = 0\) for \(n \geq N\), then \({\mathcal T} ({\mathcal S} : {\mathcal E} | n) = L\) for \(n \geq N\). Further results of this sort related to recursively transformed sequences are stated and illustrated.
Mathematics Subject Classification ID: 65B05
Mathematics Subject Classification ID: 40A25
zbMATH DE Number: 721004
zbMATH Keywords: sequence transformations
zbMATH Keywords: extrapolation methods
zbMATH Keywords: convergence acceleration
zbMATH Keywords: canonical sequences
zbMATH Keywords: difference sequence
author: Claude Brezinski
author: Michela Redivo-Zaglia
reviewed by: Peter Wynn
MaRDI profile type: Publication
full work available at URL: 0168-9274(94)00054-9
OpenAlex ID: W2058024529
cites work: Q5661062
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description: scientific article; zbMATH DE number 721004