It is possible to define convolutions in arbitrary (separable) Hilbert spaces!
More specifically, let be a separable Hilbert space, and denote by
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:
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, Funct. Anal. Appl., 51 (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
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.