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Sequence transformation

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A mathematical operator acting on a given space of sequences

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

( s n ) n ∈ N , {\displaystyle (s_{n})_{n\in \mathbb {N} },\,}

and a sequence transformation T , {\displaystyle \mathbf {T} ,} the sequence resulting from transformation by T {\displaystyle \mathbf {T} } is

T ( ( s n ) ) = ( s n ′ ) n ∈ N , {\displaystyle \mathbf {T} ((s_{n}))=(s'_{n})_{n\in \mathbb {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 , … , s n + k n ) {\displaystyle s_{n}'=T_{n}(s_{n},s_{n+1},\dots ,s_{n+k_{n}})}

for some natural number k n {\displaystyle k_{n}} for each n {\displaystyle n} and a multivariate function T n {\displaystyle T_{n}} of k n + 1 {\displaystyle k_{n}+1} variables for each n . {\displaystyle 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 {\displaystyle s_{n}} and s n ′ {\displaystyle 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 {\displaystyle T_{n}} are linear in each of their arguments for each value of n , {\displaystyle n,} for instance if

s n ′ = ∑ m = 0 k n c n , m s n + m {\displaystyle s'_{n}=\sum _{m=0}^{k_{n}}c_{n,m}s_{n+m}}

for some constants k n {\displaystyle k_{n}} and c n , 0 , … , c n , k n {\displaystyle c_{n,0},\dots ,c_{n,k_{n}}} for each n , {\displaystyle n,} then the sequence transformation T {\displaystyle \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 ) {\displaystyle (s_{n})} and the transformed sequence ( s n ′ ) {\displaystyle (s'_{n})} share the same limit ℓ {\displaystyle \ell } as n → ∞ , {\displaystyle n\rightarrow \infty ,} the transformed sequence is said to have a faster rate of convergence than the original sequence if

lim n → ∞ s n ′ − ℓ s n − ℓ = 0. {\displaystyle \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 ℓ {\displaystyle \ell } .

Examples

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

References

External links

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