Fourier inversion

In mathematics, Fourier inversion recovers a function from its Fourier transform. Several different Fourier inversion theorems exist.

Sometimes the following expression is used as the definition of the Fourier transform: $(mathcal{F}f)(t)=int_{-infty}^infty f(x), e^{-itx},dx.$

Then it is asserted that $f(x)=frac{1}{2pi}int_{-infty}^infty (mathcal{F}f)(t), e^{itx},dt.$

In this way, one recovers a function from its Fourier transform.

However, this way of stating a Fourier inversion theorem sweeps some more subtle issues under the carpet. One Fourier inversion theorem assumes that f is Lebesgue-integrable, i.e., the integral of its absolute value is finite: $int_{-infty}^inftyleft|f(x) ight|,dx

In that case, the Fourier transform is not necessarily Lebesgue-integrable. For example, the function f(x) = 1 if −a < x < a and f(x) = 0 otherwise has Fourier transform $2sin(at)/t.$

In such a case, Fourier inversion theorems usually investigate the convergence of the integral $lim_{b ightarrowinfty}frac{1}{2pi}int_{-b}^b (mathcal{F}f)(t) e^{itx},dt.$

By contrast, if we take f to be a tempered distribution -- a type of generalized function -- then its Fourier transform is another tempered distribution; and the Fourier inversion formula is then more simple to prove.

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