Theorie der Limitierungsverfahren

Zeller, Karl; Beekmann, Wolfgang

doi:10.1007/978-3-642-88470-2

Cited by 175 publications

(83 citation statements)

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“…These matrix transformations compute the elements of the transformed sequence {s ′ n } ∞ n=0 as weighted averages of the elements of the input sequence {s n } ∞ n=0 according to the general scheme s ′ n = n k=0 µ nk s k . The theoretical properties of such matrix transformations are very well understood [5][6][7][8][9][10][11][12][13]. Their main appeal lies in the fact that for the weights µ nk some necessary and sufficient conditions could be formulated which guarantee that the application of such a matrix transformation to a convergent sequence {s n } ∞ n=0 yields a transformed sequence {s ′ n } ∞ n=0 converging to the same limit s = s ∞ .…”

Section: Introduction To Convergence Acceleration and Resummationmentioning

confidence: 99%

From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions

et al. 2007

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This review is focused on the borderline region of theoretical physics and mathematics. First, we describe numerical methods for the acceleration of the convergence of series. These provide a useful toolbox for theoretical physics which has hitherto not received the attention it actually deserves. The unifying concept for convergence acceleration methods is that in many cases, one can reach much faster convergence than by adding a particular series term by term. In some cases, it is even possible to use a divergent input series, together with a suitable sequence transformation, for the construction of numerical methods that can be applied to the calculation of special functions. This review both aims to provide some practical guidance as well as a groundwork for the study of specialized literature. As a second topic, we review some recent developments in the field of Borel resummation, which is generally recognized as one of the most versatile methods for the summation of factorially divergent (perturbation) series. Here, the focus is on algorithms which make optimal use of all information contained in a finite set of perturbative coefficients. The unifying concept for the various aspects of the Borel method investigated here is given by the singularities of the Borel transform, which introduce ambiguities from a mathematical point of view and lead to different possible physical interpretations. The two most important cases are: (i) the residues at the singularities correspond to the decay width of a resonance, and (ii) the presence of the singularities indicates the existence of nonperturbative contributions which cannot be accounted for on the basis of a Borel resummation and require generalizations toward resurgent expansions. Both of these cases are illustrated by examples.

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Section: Introduction To Convergence Acceleration and Resummationmentioning

confidence: 99%

From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions

et al. 2007

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“…In particular, strong --summability and -statistical convergence are equivalent on bounded sequences (see also [7,Theorem 8]). More information on strong matrix summability can be found in [8] (for the case = 1) or [9].…”

Section: Introductionmentioning

confidence: 99%

On Convergence with respect to an Ideal and a Family of Matrices

Hardtke

2014

International Journal of Analysis

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P. Das et al. recently introduced and studied the notions of strong -summability with respect to an Orlicz function andstatistical convergence, where is a nonnegative regular matrix and is an ideal on the set of natural numbers. In this paper, we will generalise these notions by replacing with a family of matrices and with a family of Orlicz functions or moduli and study the thus obtained convergence methods. We will also give an application in Banach space theory, presenting a generalisation of Simons' sup-limsup-theorem to the newly introduced convergence methods (for the case that the filter generated by the ideal has a countable base), continuing some of the author's previous work.

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“…For basic facts on summability theory, we refer to [4,9,12] for ordinary sequences and to [1] for double sequences.…”

Section: Introductionmentioning

confidence: 99%

Summability of double sequences by weighted mean methods andTauberian conditions for convergence in Pringsheim′s sense

Móricz

Stadtmüller

2004

International Journal of Mathematics and Mathematical Sciences

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After a brief summary of Tauberian conditions for ordinary sequences of numbers, we consider summability of double sequences of real or complex numbers by weighted mean methods which are not necessarily products of related weighted mean methods in one variable. Our goal is to obtain Tauberian conditions under which convergence of a double sequence follows from its summability, where convergence is understood in Pringsheim's sense. In the case of double sequences of real numbers, we present necessary and sufficient Tauberian conditions, which are so-called one-sided conditions. Corollaries allow these Tauberian conditions to be replaced by Schmidt-type slow decrease conditions. For double sequences of complex numbers, we present necessary and sufficient so-called two-sided Tauberian conditions. In particular, these conditions are satisfied if the summable double sequence is slowly oscillating.2000 Mathematics Subject Classification: 40E05, 40B05, 40G05.

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Theorie der Limitierungsverfahren

Cited by 175 publications

References 0 publications

From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions

From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions

On Convergence with respect to an Ideal and a Family of Matrices

Summability of double sequences by weighted mean methods andTauberian conditions for convergence in Pringsheim′s sense

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