I was wondering if there are spaces (function spaces) where the functions have an integral representation, i.e. can be written as an integral involving Fourier coefficients and basis functions, akin to the Fourier transform. If so, what are they called and where may I find more information about them?
Edit: I will try to make this more specific: It is well known that a Hilbert space has an orthonormal basis such that every vector can be written as a series (which is always well defined even if the orthonormal basis is uncountable since only countably many of the Fourier coefficients are nonzero). What I'm looking for is a function space that behaves just like a Hilbert space except that instead of a series you have an integral, e.g. $L^1(\mathbf R)$ with the "basis" $\{\phi_x \mid x \in \mathbf R, \phi_x(t) = e^{itx}\}$ as used in the inverse Fourier transform.