It is possible to define convolutions in arbitrary (separable) Hilbert spaces!

More specifically, let be a separable Hilbert space, and denote by

and

two collections of elements of parametrised by a discrete set .

Assume that the system is a Riesz basis of the space and that the system is **biorthogonal** to in , i.e. we have the property that

where is the Kronecker delta, equal to 1 for , and to 0 otherwise.

Typically collections and arise as systems of eigenfunctions of some compact operator in . Namely, then are the eigenfunctions of and are the eigenfunctions of its adjoint Naturally, if is self-adjoint we have

Then we can define – and -convolutions in the following form:

and

for appropriate elements .

Then these convolutions have a number of natural properties:

- the naturally defined Fourier transforms in map these convolutions to the product of Fourier transforms. For example, defining , we have

Moreover, conversely, if a bilinear mapping satisfies , it must be given by the -convolution; - Although the bases and do not have to be orthogonal, there is a Hilbert space such that we have the Plancherel identity
- There are general families of spaces , on the Fourier transform side giving rise to further Fourier analysis in . Namely, these spaces satisfy analogues of the usual duality and interpolation relations, as well as the Hausdorff-Young inequalities with the corresponding family of subspaces of .
- The developed biorthogonal Fourier analysis can be embedded in an appropriate theory of distributions realised in suitable rigged Hilbert spaces and , with and associated to a fixed spectral set satisfying certain natural properties. These triples allow one to extend the notions of – and -convolutions and — and –Fourier transforms beyond the Hilbert space

These constructions are outlined in the subscription (~~sorry for this, need to put it on arxiv~~) paper

`Kanguzhin B., Ruzhansky M., Tokmagambetov N.`

,On convolutions in Hilbert spaces,, 51Funct. Anal. Appl.(2017), 221-224. link (eng) link (rus)

but more details can be freely seen in

`Ruzhansky M., Tokmagambetov N.`

,Convolution, Fourier analysis, and distributions generated by Riesz bases, download from arxiv

**For example**, consider the operator on the interval (0, 1) given by equipped with boundary condition for some .

The spectral properties of this have been thoroughly investigated by e.g. Titchmarsh and Cartwright:

In particular, it is easy to check that the collections

and

are the systems of eigenfunctions of and , respectively, and form Riesz bases in

In this case the abstract definition above of e.g. -convolution above yields the concrete expression

If this gives the usual convolution on the torus which can be viewed as (0,1) with periodic boundary conditions.