Convolutions in Hilbert spaces

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

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

$U:=\{u_{\xi}|\,\, u_{\xi}\in H\}_{\xi\in I}$

and

$V:=\{v_{\xi}|\,\, v_{\xi}\in H\}_{\xi\in I}$

two collections of elements of $H$ parametrised by a discrete set $I$.
Assume that the system $U$ is a Riesz basis of the space $H$ and that the system $H$ is biorthogonal to $U$ in $H$, i.e. we have the property that

$(u_{\xi},v_{\eta})_{H}=\delta_{\xi\eta},$

where $\delta_{\xi\eta}$ is the Kronecker delta, equal to 1 for $\xi=\eta$, and to 0 otherwise.

Typically collections $U$ and $V$ arise as systems of eigenfunctions of some compact operator $\mathcal L$ in $H$. Namely, then $U$ are the eigenfunctions of $\mathcal L$ and $V$ are the eigenfunctions of its adjoint $\mathcal L^*.$ Naturally, if $\mathcal L$ is self-adjoint we have $U=V.$

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

$f*_{U}g:= \sum_{\xi\in I}(f, v_\xi) (g, v_\xi) u_{\xi}$

and

$f*_{V}g:= \sum_{\xi\in I}(f, u_\xi) (g, u_\xi) v_{\xi}$

for appropriate elements $f,g\in H$.

Then these convolutions have a number of natural properties:

• the naturally defined Fourier transforms in $H$ map these convolutions to the product of Fourier transforms. For example, defining $\widehat{f}(\xi):=(f, v_{\xi})$, we have $\widehat{f*_{U} g}=\widehat{f}\,\widehat{g}.$
Moreover, conversely, if a bilinear mapping $K:H\times H\to H$ satisfies $\widehat{K(f,g)}=\widehat{f}\,\widehat{g}$, it must be given by the $U$-convolution;
• Although the bases $U$ and $V$ do not have to be orthogonal, there is a Hilbert space $l^{2}_{U}$ such that we have the Plancherel identity $(f,g)_{H}=(\widehat{f},\widehat{g})_{l^{2}_{U}}.$
• There are general families of spaces $l^{p}_{U}, 1\leq p\leq\infty$, on the Fourier transform side giving rise to further Fourier analysis in $H$. 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 $H$.
• The developed biorthogonal Fourier analysis can be embedded in an appropriate theory of distributions realised in suitable rigged Hilbert spaces $(\Phi_{U}, H, \Phi_{U}')$ and $(\Phi_{V}, H, \Phi_{V}')$, with $\Phi_{U}:=C^{\infty}_{U, \Lambda}, \Phi_{U}':=\mathcal D'_{V, \Lambda}$ and $\Phi_{V}:=C^{\infty}_{V, \Lambda}, \Phi_{V}':=\mathcal D'_{U, \Lambda},$ associated to a fixed spectral set $\Lambda$ satisfying certain natural properties. These triples allow one to extend the notions of $U$– and $V$-convolutions and $U$— and $V$–Fourier transforms beyond the Hilbert space $H.$

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 $\mathcal L$ on the interval (0, 1) given by ${\mathcal L}:= -i\frac{d}{d x}$ equipped with boundary condition $h y(0)=y(1)$ for some $h>0$.

The spectral properties of this ${\mathcal L}$ have been thoroughly investigated by e.g. Titchmarsh and Cartwright:

In particular, it is easy to check that the collections
$U=\{u_{j}(x)=h^{x} e^{2\pi i x j }, j\in {\mathbb Z}\}$
and $V=\{v_{j}(x)=h^{-x} e^{2\pi i x j }, j\in {\mathbb Z}\}$
are the systems of eigenfunctions of ${\mathcal L}$ and ${\mathcal L}^*$, respectively, and form Riesz bases in $H=L^{2}(0, 1).$
In this case the abstract definition above of e.g. $U$-convolution above yields the concrete expression

$(f*_{U}g)(x)=\int^{x}_{0}f(x-t)g(t)dt+\frac{1}{h}\int^{1}_{x}f(1+x-t)g(t)dt.$

If $h=1$ this gives the usual convolution on the torus ${\mathbb T}^1$ which can be viewed as (0,1) with periodic boundary conditions.