Convolution of test functions

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Problem 589 from Kirillov A.A., Gvishiani A.D. "Теоремы и задачи функционального анализа" [KiGv88]More problems in Subsection IV.2.3 "The Fourier Integral"

Prove that the convolution of two functions in S(Rn)S(\mathbb {R}^{n}) is also in S(Rn)S(\mathbb {R}^{n}).

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Problem 3.7.9 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.7 "Fourier Transforms"

Prove (a) of Theorem 7.8 directly (without using Fourier transforms).

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References

[KiGv88] Kirillov A.A., Gvishiani A.D. Теоремы и задачи функционального анализа. 2-е изд., Наука, 1988.
[Rud91] Rudin W. Functional Analysis. 2nd ed., McGraw-Hill, 1991.