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.
Contents
Overview
Classical examples for sequence transformations include the binomial transform, Möbius transform, Stirling transform and others.
Definitions
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
In the context of acceleration of convergence, the transformed sequence is said to converge faster than the original sequence if
where
If the mapping
for some constants
Examples
Simplest examples of (linear) sequence transformations include shifting all elements,
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
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.