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In mathematics, a sequence transformation is an operator acting on a given space of sequences (a sequence space).Sequence transformations include linear mappings such as discrete convolution with another sequence and resummation of a sequence and nonlinear mappings, more generally.
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 ...
Legendre transformation; Möbius transformation; Perspective transform (computer graphics) Sequence transform; Watershed transform (digital image processing) Wavelet transform (orthonormal) Y-Δ transform (electrical circuits)
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, the Shanks transformation is a non-linear series acceleration method to increase the rate of convergence of a sequence. This method is named after Daniel Shanks, who rediscovered this sequence transformation in 1955. It was first derived and published by R. Schmidt in 1941.
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 ...