Sequence transformation explained

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

and a sequence transformation

the sequence resulting from transformation by

T

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 .

Examples

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

• Aitken's delta-squared process
• 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