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 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. 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, and Stirling transform.

Definitions

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For a given sequence

 

and a sequence transformation   the sequence resulting from transformation by   is

 

where the elements of the transformed sequence are usually computed from some finite number of members of the original sequence, for instance

 

for some natural number   for each   and a multivariate function   of   variables for each   See for instance the binomial transform and Aitken's delta-squared process. In the simplest case the elements of the sequences, the   and  , are real or complex numbers. More generally, they may be elements of some vector space or algebra.

If the multivariate functions   are linear in each of their arguments for each value of   for instance if

 

for some constants   and   for each   then the sequence transformation   is called a linear sequence transformation. Sequence transformations that are not linear are called nonlinear sequence transformations.

In the context of series acceleration, when the original sequence   and the transformed sequence   share the same limit   as   the transformed sequence is said to have a faster rate of convergence than the original sequence if

 

If the original sequence is divergent, the sequence transformation may act as an extrapolation method to an antilimit  .

Examples

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The simplest examples of sequence transformations include shifting all elements by an integer   that does not depend on     if   and 0 otherwise, and scalar multiplication of the sequence some constant   that does not depend on     These are both examples of linear sequence transformations.

Less trivial examples include the discrete convolution of sequences with another reference sequence. A particularly basic example is the difference operator, which is convolution with the sequence   and is a discrete analog of the derivative; technically the shift operator and scalar multiplication can also be written as trivial discrete convolutions. The binomial transform and the Stirling transform are two linear transformations of a 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.

See also

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References

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