Schwartz space 4277567 209452357 2008-05-01T13:36:12Z Silly rabbit 555135 rephrase: Boundedness is a special case with p=infinity/* Properties */ In [[mathematics]], '''Schwartz space''' is the [[function space]] of rapidly decreasing functions. This space has the important property that the [[Fourier transform]] is an [[endomorphism]] on this space. This property enables one, by duality, to define the Fourier transform for elements in the dual space of <math>\mathcal{S}</math>, that is, for [[tempered distributions]]. Schwartz space is named in honour of [[Laurent Schwartz]]. A function in the Schwartz space is sometimes called a ''Schwartz function''. [[Image:Gaussian 2d.png|right|thumb|250px|A two-dimensional [[Gaussian function]] is an example of a rapidly decreasing function.]] ==Definition== The Schwartz space or '''space of rapidly decreasing functions''' <math>\mathcal{S}</math> on '''R'''<sup>n</sup> is the function space :<math> \mathcal{S} \left(\mathbb{R}^n\right) = \{ f \in C^\infty(\mathbb{R}^n) \mid ||f||_{\alpha,\beta} < \infty\, \forall \, \alpha, \beta \}, </math> where &alpha;, &beta; are [[multi-index|multi-indices]], C<sup>&infin;</sup>('''R'''<sup>n</sup>) is the set of smooth functions from '''R'''<sup>n</sup> to '''C''', and :<math>||f||_{\alpha,\beta}=||x^\alpha D^\beta f||_\infty\,.</math> Here, <math>||\cdot||_\infty</math> is the [[supremum norm]], and we use multi-index notation. When the dimension ''n'' is clear, it is convenient to write <math>\mathcal{S}=\mathcal{S}(\mathbb{R}^n)</math>. ==Examples of functions in S== * If ''i'' is a multi-index, and ''a'' is a positive real number, then :<math>x^i e^{-a x^2} \in \mathcal{S} (\mathbb{R}).</math> * Any smooth function ''f'' with compact support is in <math>\mathcal{S}</math>. This is clear since any derivative of ''f'' is continuous, so (''x''<sup>&alpha;</sup> D<sup>&beta;</sup>) ''f'' has a maximum in '''R'''<sup>n</sup>. ==Properties== * <math>\mathcal{S}</math> is a [[Fréchet space]] over complex numbers. In other words, <math>\mathcal{S}</math> is closed under point-wise addition and under multiplication by a complex scalar. * Using [[Leibniz_integral_rule|Leibniz' rule]], it follows that <math>\mathcal{S}</math> is also closed under point-wise multiplication; if <math>f,g \in \mathcal{S}</math>, then <math>fg: x\mapsto f(x)g(x)</math> is also in <math>\mathcal{S}</math>. * For any 1 &le; ''p'' &le; &infin;, we have <math>\mathcal{S}\subset L^p,</math> where ''L''<sup>p</sup>('''R'''<sup>n</sup>) is the space of [[Lp space|''p''-integrable functions]] on '''R'''<sup>n</sup>. In particular, functions in <math>\mathcal{S}</math> are bounded (Reed & Simon 1980). * The Fourier transform is a linear isomorphism <math>\mathcal{S} \to \mathcal{S}</math>. ==References== * L. Hörmander, ''The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis)'', 2nd ed, Springer-Verlag, 1990. * M. Reed, B. Simon, ''Methods of Modern Mathematical Physics: Functional Analysis I, Revised and enlarged edition'', Academic Press, 1980. {{planetmath|id=4444|title=Space of rapidly decreasing functions }} [[Category:Topological vector spaces]] [[Category:Smooth functions]] [[Category:Fourier analysis]] [[fr:Espace de Schwartz]] [[it:Spazio di Schwartz]]