Summation formulas for integral transforms other than the Fourier

It seems to me that the Fourier transform harbours multiple useful results that allow one to sum an infinite series. The Poisson Summation Formula and Parseval's theorem. Also, Plancherel's theorem allows one to find the value of certain definite integrals.

These methods are all based on the Fourier transform. Yet it also appears to me that there are fewer theorems and results pertaining to infinite sums and integrals for other integral transforms, like the Abel transform, the Hermite transform, and the Laguerre transform.

Question: are there also summation formulas for integral transforms other than the Fourier transform?

Solution 1:

The Hermite transform says that the $ H_n$ form an orthogonal basis of $L^2(\Bbb{R},e^{-x^2})$ the Hilbert space with inner product $( f,g)=\int_{-\infty}^\infty f(x)\overline{g(x)}e^{-x^2}dx$ so that $$f =H_n \frac{( f,H_n)}{(H_n,H_n)}, \qquad (f,f)= \sum_n \frac{( f ,f)}{(H_n,H_n)} $$

This is similar to Parseval for the Fourier series where the $e^{2i\pi nx}$ form an orthogonal basis of $L^2[0,1]$ with inner product $\langle f,g\rangle=\int_0^1 f(x)\overline{g(x)}dx$.

Parseval for the Fourier transform is a bit more general, it says that $f\to \mathcal{F}(f)$ is an unitary operator from $A=L^2(\Bbb{R})$ to $B= L^2(\Bbb{R})$, ie. $( f,f)_A=(\mathcal{F}(f),\mathcal{F}(f))_B$.

The Hermite transform is unitary from $L^2(\Bbb{R},e^{-x^2})$ to $\ell^2(\Bbb{Z}_{\ge 0},(c_n))$ with $c_n=\frac1{(H_n,H_n)}$