Fourier inversion theorem

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

Sometimes the following identity 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; it may be only "conditionally 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, the integral in the Fourier inversion theorem above must be taken to be an improper integral (Cauchy principal value)

:$$lim_{b ightarrowinfty}frac{1}{2pi}int_{-b}^b (mathcal{F}f)(t) e^{itx},dt

rather than a Lebesgue integral.

By contrast, if we take "f" to be a tempered distribution -- a sort of generalized function -- then its Fourier transform is a function of the same sort: another tempered distribution; and the Fourier inversion formula is more simply proved.

Fourier transforms of quadratically integrable functions

Via the Plancherel theorem, one can also define the Fourier transform of a quadratically integrable function, i.e., one satisfying

:$$int_{-infty}^inftyleft|f(x) ight|^2,dx

Then the Fourier transform is another quadratically integrable function.

In case "f" is a quadratically integrable periodic function on the intervalthen it has a Fourier series whose coefficients are

:$$widehat{f}(n)=frac{1}{2pi}int_{-pi}^pi f(x),e^{-inx},dx.

The Fourier inversion theorem might then say that

:$$sum_{n=-infty}^{infty} widehat{f}(n),e^{inx}=f(x).

What kind of convergence is right? "Convergence in mean square" can be proved fairly easily:

:$$lim_{N ightarrowinfty}int_{-pi}^pileft|f(x)-sum_{n=-N}^{N} widehat{f}(n),e^{inx} ight|^2,dx=0.

What about convergence almost everywhere? That would say that if "f" is quadratically integrable, then for "almost every" value of "x" between 0 and 2π we have

:$$f(x)=lim_{N ightarrowinfty}sum_{n=-N}^{N} widehat{f}(n),e^{inx}.

This was not proved until 1966 in (Carleson, 1966).

For strictly finitary discrete Fourier transforms, these delicate questions of convergence are avoided.

References

*Lennart Carleson (1966). On the convergence and growth of partial sums of Fourier series. "Acta Math." 116, 135&ndash;157.