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.
Classical examples for sequence transformations include the binomial transform, Möbius transform, Stirling transform and others.
For a given sequence
S=\{sn\}n\in\N,
the transformed sequence is
T(S)=S'=\{s'n\}n\in\N,
where the members of the transformed sequence are usually computed from some finite number of members of the original sequence, i.e.
sn'=T(sn,sn+1,...,sn+k)
for some
k
n
sn
s'n
In the context of acceleration of convergence, the transformed sequence is said to converge faster than the original sequence if
\limn\toinfty
s'n-\ell | |
sn-\ell |
=0
where
\ell
S
\ell
If the mapping
T
s'n=\sum
k | |
m=0 |
cmsn+m
for some constants
c0,...,ck
T
Simplest examples of (linear) sequence transformations include shifting all elements,
s'n=sn+k
A less trivial example would be the discrete convolution with a fixed sequence. A particularly basic form is the difference operator, which is convolution with the sequence
(-1,1,0,\ldots),
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.