Sequence transformation

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 convolution with another sequence, and resummation of a sequence and, more generally, are commonly used for series acceleration, that is, for improving the rate of convergence of a slowly convergent sequence or series. Sequence transformations are also commonly used to compute the antilimit of a divergent series numerically, and are used in conjunction with extrapolation methods.


Classical examples for sequence transformations include the binomial transform, Möbius transform, Stirling transform and others.


For a given sequence


the transformed sequence is


where the members of the transformed sequence are usually computed from some finite number of members of the original sequence, i.e.


for some   which often depends on   (cf. e.g. Binomial transform). In the simplest case, the   and the   are real or complex numbers. More generally, they may be elements of some vector space or algebra.

In the context of acceleration of convergence, the transformed sequence is said to converge faster than the original sequence if


where   is the limit of  , assumed to be convergent. In this case, convergence acceleration is obtained. If the original sequence is divergent, the sequence transformation acts as extrapolation method to the antilimit  .

If the mapping   is linear in each of its arguments, i.e., for


for some constants   (which may depend on n), the sequence transformation   is called a linear sequence transformation. Sequence transformations that are not linear are called nonlinear sequence transformations.


Simplest examples of (linear) sequence transformations include shifting all elements,   (resp. = 0 if n + k < 0) for a fixed k, and scalar multiplication of the sequence.

A little less trivial generalization would be the discrete convolution with a fixed sequence. A particularly basic form is the difference operator, which is convolution with the sequence   and is a discrete analog of the derivative. The binomial transform is another linear transformation of a still more general type.

An example of a nonlinear sequence transformation is Aitken's delta-squared process, used to improve the rate of convergence of a slowly convergent sequence. An extended form of this is the Shanks transformation. The Möbius transform is also a nonlinear transformation, only possible for integer sequences.

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