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| ==Overview==
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| Classical examples for sequence transformations include the [[binomial transform]], [[Möbius transform]], [[Stirling transform]] and others.
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| ==Definitions==
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| For a given sequence
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| :<math>S=\{ s_n \}_{n\in\N},\,</math>
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| the '''transformed sequence''' is
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| :<math>\mathbf{T}(S)=S'=\{ s'_n \}_{n\in\N},\,</math>
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| where the members of the transformed sequence are usually computed from some finite number of members of the original sequence, i.e.
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| :<math>s_n' = T(s_n,s_{n+1},\dots,s_{n+k})</math>
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| for some <math>k</math> which often depends on <math>n</math> (cf. e.g. [[Binomial transform]]). In the simplest case, the <math>s_n</math> and the <math>s'_n</math> are [[real number|real]] or [[complex number]]s. More generally, they may be elements of some [[vector space]] or [[algebra]]. | |
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| In the context of acceleration of convergence, the transformed sequence is said to '''converge faster''' than the original sequence if
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| :<math>\lim_{n\to\infty} \frac{s'_n-\ell}{s_n-\ell} = 0</math> | |
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| where <math>\ell</math> is the limit of <math>S</math>, assumed to be convergent. In this case, [[convergence acceleration]] is obtained. If the original sequence is [[Divergent sequence|divergent]], the sequence transformation acts as [[extrapolation method]] to the antilimit <math>\ell</math>.
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| If the mapping <math>T</math> is [[linear]] in each of its arguments, i.e., for
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| :<math>s'_n=\sum_{m=0}^{k} c_m s_{n+m}</math> | |
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| for some constants <math>c_0,\dots,c_k</math> (which may depend on ''n''), the sequence transformation <math>\mathbf{T}</math> is called a '''linear sequence transformation'''. Sequence transformations that are not linear are called [[nonlinear sequence transformation]]s.
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| ==Examples==
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| Simplest examples of (linear) sequence transformations include shifting all elements, <math>s'_n = s_{n+k}</math> (resp. = 0 if ''n'' + ''k'' < 0) for a fixed ''k'', and [[scalar multiplication]] of the sequence.
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| A little less trivial generalization would be the [[convolution#Discrete convolution|discrete convolution]] with a fixed sequence. A particularly basic form is the [[difference operator]], which is convolution with the sequence <math>(-1,1,0,\ldots),</math> and is a discrete analog of the derivative. The [[binomial transform]] is another linear transformation of a still more general type.
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| 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 sequence]]s.
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| ==See also ==
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| * [[Series acceleration]]
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| * [[Minimum polynomial extrapolation]]
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| ==References==
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| <references/> | |
| *Hugh J. Hamilton, "[http://www.ams.org/bull/1947-53-08/S0002-9904-1947-08882-0/S0002-9904-1947-08882-0.pdf Mertens' Theorem and Sequence Transformations]", AMS (1947)
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| ==External links==
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| * [http://oeis.org/transforms.html Transformations of Integer Sequences], a subpage of the [[On-Line Encyclopedia of Integer Sequences]]
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| [[Category:Mathematical series]]
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| [[Category:Asymptotic analysis]]
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| [[Category:Perturbation theory]]
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| [[es:Transformación de sucesiones]]
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| [[de:Folgentransformation]]
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| [[fr:Delta-2]]
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| [[ru:Преобразование последовательностей]]
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