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

For a given sequence

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

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

are linear for instance if

is called a linear sequence transformation. Sequence transformations that are not linear are called nonlinear sequence transformations.

In the context of series acceleration 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

and 0 otherwise, and scalar multiplication 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 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

• Aitken's delta-squared process

• Anderson acceleration

• Minimum polynomial extrapolation

• Richardson extrapolation

• Series acceleration

• Steffensen's method

References

• Hugh J. Hamilton, "Mertens' Theorem and Sequence Transformations", AMS (1947)

External links

• Transformations of Integer Sequences, a subpage of the On-Line Encyclopedia of Integer Sequences