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Sequence transformations include linear mappings such as discrete convolution with another sequence and resummation of a sequence and nonlinear mappings, more generally. They are commonly used for series acceleration , that is, for improving the rate of convergence of a slowly convergent sequence or series .
Two classical techniques for series acceleration are Euler's transformation of series [1] and Kummer's transformation of series. [2] A variety of much more rapidly convergent and special-case tools have been developed in the 20th century, including Richardson extrapolation, introduced by Lewis Fry Richardson in the early 20th century but also known and used by Katahiro Takebe in 1722; the ...
A Möbius transformation can be composed as a sequence of simple transformations. ... Jonathan (2008), "Möbius Transformations Revealed" (PDF), Notices of the AMS ...
These transformations typically involve integral formulas applied to a sequence generating function (see integral transformations) or weighted sums over the higher-order derivatives of these functions (see derivative transformations). Given a sequence, {} =, the ordinary generating function (OGF) of the sequence, denoted (), and the exponential ...
Download as PDF; Printable version ... Kummer's transformation of series is a method used to accelerate the ... is always also a sequence going to zero and the series ...
In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is the result of applying the binomial transform to the sequence associated with its ordinary generating function.
In numerical analysis, Aitken's delta-squared process or Aitken extrapolation is a series acceleration method used for accelerating the rate of convergence of a sequence. It is named after Alexander Aitken, who introduced this method in 1926. [1] It is most useful for accelerating the convergence of a sequence that is converging linearly.
In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation, named after Niels Henrik Abel who introduced it in 1826. [1]