- Schwartz space
In
mathematics , Schwartz space is thefunction space of rapidly decreasing functions. This space has the important property that theFourier transform is anendomorphism on this space. This property enables one, by duality, to define the Fourier transform for elements in the dual space of mathcal{S}, that is, fortempered distributions . Schwartz space is named in honour ofLaurent Schwartz . A function in the Schwartz space is sometimes called a "Schwartz function".Definition
The Schwartz space or space of rapidly decreasing functions mathcal{S} on Rn is the function space
:mathcal{S} left(mathbb{R}^n ight) = { f in C^infty(mathbb{R}^n) mid ||f||_{alpha,eta} < infty, forall , alpha, eta },
where α, β are multi-indices, C∞(Rn) is the set of smooth functions from Rn to C, and
:f||_{alpha,eta}=||x^alpha D^eta f||_infty,.
Here, cdot||_infty is the
supremum norm , and we use multi-index notation. When the dimension "n" is clear, it is convenient to write mathcal{S}=mathcal{S}(mathbb{R}^n).Examples of functions in S
* If "i" is a multi-index, and "a" is a positive real number, then:x^i e^{-a x^2} in mathcal{S} (mathbb{R}).
* Any smooth function "f" with compact support is in mathcal{S}. This is clear since any derivative of "f" is continuous, so ("x"α Dβ) "f" has a maximum in Rn.
Properties
* mathcal{S} is a
Fréchet space over the complex numbers.* Using Leibniz' rule, it follows that mathcal{S} is also closed under point-wise multiplication; if f,g in mathcal{S}, then fg: xmapsto f(x)g(x) is also in mathcal{S}.
* For any 1 ≤ "p" ≤ ∞, we have mathcal{S}subset L^p, where "L"p(Rn) is the space of "p"-integrable functions on Rn. In particular, functions in mathcal{S} are bounded (Reed & Simon 1980).
* The Fourier transform is a linear isomorphism mathcal{S} o mathcal{S}.
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.
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