Answer to Question #97529 in Functional Analysis for Rachel

Question #97529
Let TES^1. Prove that for all φES, (T*φ)^ = (2π)^(n/2)(φ-hat)(T-hat).
1
Expert's answer
2019-11-04T09:24:22-0500

"T\\in S(given)"

To prove-for all "\\psi\\in S,F(T*\\psi)=(2\u03c0)^{n\/2}.F(\\psi).F(T)"

"F(T(t))= fourier" transform of T(t)

"(T*\\psi)(t)=\\int_{-\\infty} ^{\\infty}T(x)\\psi(t-x)dx"

"F(T(t))=(2\u03c0)^{-1\/2}(\\int_{-\\infty}^{\\infty}T(t)exp(-iwt)dt"

"F((T*\\psi)(t))=(2\u03c0)^{-1\/2}(\\int_{-\\infty}^{\\infty} \n (T*\\psi)(t) exp(-iwt)dt"

"F((T*\\psi)(t))=(2\u03c0)^{-1\/2}(\\int_{-\\infty}^{\\infty} (\\int_{-\\infty} ^{\\infty}T(x)\\psi(t-x)exp(-iwt)dx)dt"

We can change the order of the equation and separate T(x) as it will be a constant when integrated with respect to t.

"F((T*\\psi)(t))=(\\int_{-\\infty}^{\\infty} T(x) (2\u03c0)^{-1\/2} (\\int_{-\\infty} ^{\\infty}\\psi(t-x)exp(-iwt)dt)dx"

"(2\u03c0)^{-1\/2} (\\int_{-\\infty} ^{\\infty}\\psi(t-x)exp(-iwt)dt)=F(\\psi(t-x))"

"F((T*\\psi)(t))=(\\int_{-\\infty}^{\\infty} T(x) F(\\psi(t-x))dx" ----(i)

By properties of Fourier Transform,

"F(\\psi(t-x))=exp(-iwt)F(\\psi(x))"

Using this in equation (i)

"F((T*\\psi)(t))=\\int_{-\\infty}^{\\infty} T(x) exp(-iwt)F(\\psi(t)dx"

"F((T*\\psi)(t))=(2\u03c0) ^{1\/2}F(\\psi(t).(2\u03c0)^{-1\/2 }. \\int_{-\\infty}^{\\infty} T(x) exp(-iwt)dx"

"F((T*\\psi)(t))=(2\u03c0) ^{1\/2}F(\\psi(t).F(T(t))"

So,

"F(T*\\psi)=(2\u03c0)^{n\/2}.F(\\psi).F(T)" where "n=1"

(Proved)






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