The Fourier transform is a integral transform with kernel $e^ {−2πiξx}$. The Fourier transform is unitary in that it preserves the $L2 $ norm.
Is there a general way to show or guess that a kernel is unitary? For example, if I arbitrarily came up with a kernel $K(ξ,x) = ξ^x$ or $K(ξ,x) = ξ+x$ how would I determine if it is unitary or not. Any information about the 'space' of unitary kernels would be much appreciated.
Let's denote the integral transform of $f$ by $F(\xi) := \int\mathrm{d}x\, f(x)K(\xi,x)$. Then, unitarity is translated $\langle F,F \rangle = \langle f,f \rangle$. If we consider the standard inner product of $L^2$, the RHS is $\langle f,f \rangle = \int\mathrm{d}x\, |f(x)|^2$, when the LHS is given by $$ \langle F,F \rangle = \int\mathrm{d}\xi\, |F(\xi)|^2 = \int\mathrm{d}\xi \int\mathrm{d}x_1\int \mathrm{d}x_2\, \overline{f(x_1)}f(x_2) \, \overline{K(\xi,x_1)}K(\xi,x_2), $$ so that the condition of unitarity is written as $$ \delta(x_1-x_2) = \int\mathrm{d}\xi\; \overline{K(\xi,x_1)}K(\xi,x_2), $$ where $\delta$ denotes the Dirac delta function.