** Fourier Transform in **

DefinitionFor , we define

Note that (akaMinkowski’s inequality).

DefinitionA smooth function is said to be aSchwartz function(this is Laurent Schwartz, not Hermann Schwarz of the C-S inequality) if

where and . The space of Schwartz function is denoted .

PropositionSuppose , then

- If , then .
- If then .
- If , where , then .
- If then .
- If with , then (again, Minkowski’s inequality) and .
- If then .
- If then .

*Proof:* For assertion (3), using the change of variables ,

For assertion (5), using the change of variables and Fubini

For the assertion (7), note that . The remaining assertions are left as an exercise to the reader.

Remark

- Using , we can extend assertions (1) through (5) to general .
- From assertions (6) and (7) we see that if , then .
- From assertion (3) we see that if , then . In particular, any reflection/rotation symmetry of is inherited by .

LemmaLet be a complex matrix with positive definite real part. Then

*Note:* The fact that is indeed invertible is left as an exercise to the reader. *Proof:* By analytic continuation it suffices to consider the case where is real (this method also reveals which branch of one should use). Let then

Thus, the whole problem is to show

Note that

with the last equality by the fundamental theorem of calculus, and thus we are left to deal with the case . Breaking into eigenvalues and eigenvectors, we are just left to show that

which is a classical computation from multivariable calculus.

RemarkFrom this result it follows that the Fourier transform of is itself.

Theorem (Fourier Inversion)If , then(note that the integral on the right hand side is absolutely convergent since ). Equivalently,

We call this “inverse Fourier transform” .

*Proof:* Let . By dominated convergence theorem, is equal to the right hand side above. On the other hand, Fubini’s theorem gives

which converges to as because is continuous.

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