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space of rapidly decreasing functions∗
matte†
2013-03-21 16:14:40
The function space of rapidly decreasing functions S 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 S, that is, for tempered distributions.
Definition The space of rapidly decreasing functions on Rn is the function
space
S(Rn ) = {f ∈ C ∞ (Rn ) | sup | ||f ||α,β < ∞ for all multi-indices α, β},
x∈Rn
where C ∞ (Rn ) is the set of smooth functions from Rn to C, and
||f ||α,β = ||xα Dβ f ||∞ .
Here, || · ||∞ is the supremum norm, and we use multi-index notation. When
the dimension n is clear, it is convenient to write S = S(Rn ). The space S is
also called the Schwartz space, after Laurent Schwartz (1915-2002) [?].
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Examples of functions in S
1. If i is a multi-index, and a is a positive real number, then
xi exp{−ax2 } ∈ S.
2. Any smooth function with compact support f is in S. This is clear since
any derivative of f is continuous, so xα Dβ f has a maximum in Rn .
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Properties
1. S is a complex vector space. In other words, S is closed under point-wise
addition and under multiplication by a complex scalar.
∗ hSpaceOfRapidlyDecreasingFunctionsi
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2. Using Leibniz’ rule, it follows that S is also closed under point-wise multiplication; if f, g ∈ S, then f g : x 7→ f (x)g(x) is also in S.
3. For any 1 ≤ p ≤ ∞, we have [?]
S ⊂ Lp ,
and if p < ∞, then S is also dense in Lp .
4. The Fourier transform is a linear isomorphism S → S.
References
[1] L. Hörmander, The Analysis of Linear Partial Differential Operators I, (Distribution theory and Fourier Analysis), 2nd ed, Springer-Verlag, 1990.
[2] The MacTutor History of Mathematics archive, Laurent Schwartz
[3] M. Reed, B. Simon, Methods of Modern Mathematical Physics: Functional
Analysis I, Revised and enlarged edition, Academic Press, 1980.
[4] Wikipedia, Tempered distributions
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