, a sequence transformation
is an operator
acting on a given space of sequences
. 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
. 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
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
< 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. 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. The Möbius transform is also a nonlinear transformation, only possible for integer sequences.