Let be the set of non-negative integers, and for any , let be the n-fold Cartesian product.
The Schwartz space or space of rapidly decreasing functions on is the function spacewhere is the function space of smooth functions from into , and Here, denotes the supremum, and we used multi-index notation, i.e. and .
To put common language to this definition, one could consider a rapidly decreasing function as essentially a function f(x) such that f(x), f ′(x), f ′′(x), ... all exist everywhere on R and go to zero as x→ ±∞ faster than any reciprocal power of x. In particular, S(Rn, C) is a subspace of the function space C∞(Rn, C) of smooth functions from Rn into C.
- If is a multi-index, and a is a positive real number, then
- Any smooth function f with compact support is in S(Rn). This is clear since any derivative of f is continuous and supported in the support of f, so ( has a maximum in Rn by the extreme value theorem.
- Because the Schwartz space is a vector space, any polynomial can be multiplied by a factor for a real constant, to give an element of the Schwartz space. In particular, there is an embedding of polynomials inside a Schwartz space.
Relation of Schwartz spaces with other topological vector spaces
- If 1 ≤ p ≤ ∞, then 𝒮(Rn) ⊂ Lp(Rn).
- If 1 ≤ p < ∞, then 𝒮(Rn) is dense in Lp(Rn).
- The space of all bump functions, C∞
c(Rn), is included in 𝒮(Rn).
Sources
- Hörmander, L. (1990). The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis) (2nd ed.). Berlin: Springer-Verlag. ISBN 3-540-52343-X.
- Reed, M.; Simon, B. (1980). Methods of Modern Mathematical Physics: Functional Analysis I (Revised and enlarged ed.). San Diego: Academic Press. ISBN 0-12-585050-6.
- Stein, Elias M.; Shakarchi, Rami (2003). Fourier Analysis: An Introduction (Princeton Lectures in Analysis I). Princeton: Princeton University Press. ISBN 0-691-11384-X.
- Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.
This article incorporates material from Space of rapidly decreasing functions on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.