Sequence transformation

For a given sequence :(s_n)_{n\in\N},\, and a sequence transformation \mathbf{T}, the sequence resulting from transformation by \mathbf{T} is :\mathbf{T}( ( s_n ) ) = ( s'_n )_{n\in\N}, where the elements of the transformed sequence are usually computed from some finite number of members of the original sequence, for instance :s_n' = T_n(s_n,s_{n+1},\dots,s_{n+k_n}) for some natural number k_n for each n and a multivariate function T_n of k_n + 1 variables for each n. See for instance the binomial transform and Aitken's delta-squared process. In the simplest case the elements of the sequences, the s_n and s'_n, are real or complex numbers. More generally, they may be elements of some vector space or algebra. If the multivariate functions T_n are linear in each of their arguments for each value of n, for instance if :s'_n=\sum_{m=0}^{k_n} c_{n,m} s_{n+m} for some constants k_n and c_{n,0},\dots,c_{n,k_n} for each n, then the sequence transformation \mathbf{T} 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 (s_n) and the transformed sequence (s'_n) share the same limit \ell as n \rightarrow \infty, the transformed sequence is said to have a faster rate of convergence than the original sequence if :\lim_{n\to\infty} \frac{s'_n-\ell}{s_n-\ell} = 0. If the original sequence is divergent, the sequence transformation may act as an extrapolation method to an antilimit \ell. ==Examples==

The simplest examples of sequence transformations include shifting all elements by an integer k that does not depend on n, s'_n = s_{n+k} if n + k \geq 0 and 0 otherwise, and scalar multiplication of the sequence some constant c that does not depend on n, s'_n = c s_{n}. 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 (-1,1,0,\ldots) 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 ==