We study infinite matrices $A$ indexed by a discrete group $G$ that are dominated by a convolution operator in the sense that $|(Ac)(x)| \leq (a \ast |c|)(x)$ for $x\in G$ and some $a\in \ell ^1(G)$. This class of "convolution-dominated" matrices forms a Banach-*-algebra contained in the algebra of bounded operators on $\ell ^2(G)$. Our main result shows that the inverse of a convolution-dominated matrix is again convolution-dominated, provided that $G$ is amenable and rigidly symmetric. For abelian groups this result goes back to Gohberg, Baskakov, and others, for non-abelian groups completely different techniques are required, such as generalized
$L^1$-algebras and the symmetry of group algebras.
For a locally compact group $G$ we consider the algebra $CD(G)$ of convolution dominated operators on $L^{2}(G)$: An operator $A:L^2(G)\to L^2(G)$ is called convolution dominated if there exists $a\in L^1(G)$ such that for all $f \in L^2(G)$ $ |Af(x)| \leq a * |f| (x)$, for almost all $x \in G$. In the case of discrete groups those operators can be dealt with quite sufficiently if the group in question is rigidly symmetric. For non-discrete groups we investigate the subalgebra of regular convolution dominated operators $CD_{reg}(G)$. For amenable $G$ which is rigidly symmetric as a discrete gr…
We show that sampling or interpolation formulas in reproducing kernel Hilbert spaces can be obtained by reproducing kernels whose dual systems form molecules, ensuring that the size profile of a function is fully reflected by the size profile of its sampled values. The main tool is a local holomorphic calculus for convolution-dominated operators, valid for groups with possibly non-polynomial growth. Applied to the matrix coefficients of a group representation, our methods improve on classical results on atomic decompositions and bridge a gap between abstract and concrete methods.
If $G$ is a locally compact group, $CD(G)$ the algebra of convolution dominated operators on $L^2(G)$ then an important question is: Is $\mathbb{C}1+CD(G)$ (respectively $CD(G)$ if $G$ is discrete) inverse-closed in the bounded operators on $L^2(G)$? In this note we answer this question in the affirmative provided $G$ is such that one of the following properties is fulfilled (1) There is a discrete, rigidly symmetric, and amenable subgroup $H\subset G$ and a (measurable) relatively compact neighbourhood of the identity $U$ invariant under conjugation by elements of $H$ such that $\{hU\;:\;h\in…
We study the bounded endomorphisms of $\ell_{N}^2(G)=\ell^2(G)\times \dots \times\ell^2(G)$ that commute with translations, where $G$ is a discrete abelian group. It is shown that they form a C*-algebra isomorphic to the C*-algebra of $N\times N$ matrices with entries in $L^\infty(\widehat{G})$, where $\widehat{G}$ is the dual space of $G$. Characterizations of when these endomorphisms are invertible, and expressions for their norms and for the norms of their inverses, are given. These results allow us to study Riesz systems that arise from the action of $ G $ on a finite set of elements of a …
The Fractional Fourier Transform is a ubiquitous signal processing tool in basic and applied sciences. The Fractional Fourier Transform generalizes every property and application of the Fourier Transform. Despite the practical importance of the discrete fractional Fourier transform, its applications in digital communications have been elusive. The convolution property of the discrete Fourier transform plays a vital role in designing multi-carrier modulation systems. Here we report a closed-form affine discrete fractional Fourier transform and we show the circular convolution property for it. T…
This work is focused on establishing sufficient conditions to guarantee the well-posedness of the following nonlinear fractional semidiscrete model \begin{equation*} \begin{cases} \mathbb D^β_t u(n,t)= B u(n,t) + f(n-ct,u(n,t)),\, &n\in\mathbb{Z}, \;t>0, u(n,0)=φ(n),\; &n\in\mathbb{Z}, \end{cases} \end{equation*} under the assumptions that $β\in (0,1]$, $c>0$ some constant, $B$ is a discrete convolution operator with kernel $b\in\ell^1(\Z)$, which is the infinitesimal generator of the Markovian $C_0$-semigroup and suitable nonlinearity $f$. We present results concerning the existence and uniqu…
We establish a necessary and sufficient criterion for the Fredholmness of a general locally compact band-dominated operator $A$ on $L^p(R)$ and solve the long-standing problem of computing its Fredholm index in terms of the limit operators of $A$. The results are applied to operators of convolution type with almost periodic symbol.
A regular sampling theory in a multiply generated unitary invariant subspace of a separable Hilbert space $\mathcal{H}$ is proposed. This subspace is associated to a unitary representation of a countable discrete abelian group $G$ on $\mathcal{H}$. The samples are defined by means of a filtering process which generalizes the usual sampling settings. The multiply generated setting allows to consider some examples where the group $G$ is non-abelian as, for instance, crystallographic groups. Finally, it is worth to mention that classical average or pointwise sampling in shift-invariant subspaces …
In this paper we develop constructive invertibility conditions for the twisted convolution. Our approach is based on splitting the twisted convolution with rational parameters into a finite number of weighted convolutions, which can be interpreted as another twisted convolution on a finite cyclic group. In analogy with the twisted convolution of finite discrete signals, we derive an anti-homomorphism between the sequence space and a suitable matrix algebra which preserves the algebraic structure. In this way, the problem reduces to the analysis of finite matrices whose entries are sequences su…
We introduce multi-scale filtering by the way of certain double convolution systems. We prove stability theorems for these systems and make connections with function theory in the poly-disc. Finally, we compare the framework developed here with the white noise space framework, within which a similar class of double convolution systems has been defined earlier.
This paper focuses on the spectral properties of a bounded self-adjoint operator in $L_2(\mathds R^d)$ being the sum of a convolution operator with an integrable convolution kernel and an operator of multiplication by a continuous potential converging to zero at infinity. We study both the essential and the discrete spectra of this operator. It is shown that the essential spectrum of the sum is the union of the essential spectrum of the convolution operator and the image of the potential. We then provide a number of sufficient conditions for the existence of discrete spectrum and obtain lower …
Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a C*-bialgebra, the noncommutative counterpart of locally compact semigroup. On locally compact quantum groups we obtain a bijective correspondence between such convolution semigroups and a class of C_0-semigroups of maps which we characterise. On C*-bialgebras of discrete type we show that all weakly continuous convolution semigroups of s…
We put together the ingredients for an efficient operator calculus based on Krawtchouk polynomials, including Krawtchouk transforms and corresponding convolution structure which provide an inherently discrete alternative to Fourier analysis. In this paper, we present the theoretical aspects and some basic examples.
For a locally compact group G, the convolution product on the space N(L^p(G)) of nuclear operators was defined by Neufang. We study homological properties of the convolution algebra N(L^p(G)) and relate them with some properties of the group G, such as compactness, finiteness, discreteness, and amenability.
Let $G$ be a discrete group, let $p\ge1$, and let $ω$ be a weight on $G$. Using the approach from [9], we provide sufficient conditions on a weight $ω$ for $\ell^p(G,ω)$ to be a Banach algebra admitting a norm-controlled inversion in the reduced C$^*$-algebra of $G$, namely $C^*_r(G)$. We show that our results can be applied to various cases including locally finite groups as well as finitely generated groups of polynomial or intermediate growth and a natural class of weights on them. These weights are of the form of polynomial or certain subexponential functions. We also consider the non-disc…
Motivated by the existence problem of Fourier frames on fractal measures, we introduce Bessel and frame measures for a given finite measure on $\br^d$, as extensions of the notions of Bessel and frame spectra that correspond to bases of exponential functions. Not every finite compactly supported Borel measure admits frame measures. We present a general way of constructing Bessel/frame measures for a given measure. The idea is that if a convolution of two measures admits a Bessel measure then one can use the Fourier transform of one of the measures in the convolution as a weight for the Bessel …
We obtain weak type (1,1) estimates for the inverses of truncated discrete rough Hilbert transform. We include an ex- ample showing that our result is sharp. One of the ingredients of the proof are regularity estimates for convolution of singular measure associated with the sequence $[m^α]$.
This paper introduces a notion of gradient and an infimal-convolution operator that extend properties of solutions of Hamilton Jacobi equations to more general spaces, in particular to graphs. As a main application, the hypercontractivity of this class of infimal-convolution operators is connected to some discrete version of the log-Sobolev inequality and to a discrete version of Talagrand's transport inequality.
In this paper we introduce a new way of deforming convolution algebras and Fourier algebras on locally compact groups. We demonstrate that this new deformation allows us to reveal some informations of the underlying groups by examinining Banach algebra properties of deformed algebras. More precisely, we focus on representability as an operator algebra of deformed convolution algebras on compact connected Lie groups with connection to the real dimension of the underlying group. Similary, we investigate complete representability as an operator algebra of deformed Fourier algebras on some finitel…
Let $(\cX, ρ)$ be a discrete metric space. We suppose that the group $\sZ^n$ acts freely on $X$ and that the number of orbits of $X$ with respect to this action is finite. Then we call $X$ a $\sZ^n$-periodic discrete metric space. We examine the Fredholm property and essential spectra of band-dominated operators on $l^p(X)$ where $X$ is a $\sZ^n$-periodic discrete metric space. Our approach is based on the theory of band-dominated operators on $\sZ^n$ and their limit operators. In case $X$ is the set of vertices of a combinatorial graph, the graph structure defines a Schrödinger operator on $l…
In this paper we study the regularity properties of two maximal operators of convolution type: the heat flow maximal operator (associated to the Gauss kernel) and the Poisson maximal operator (associated to the Poisson kernel). In dimension $d=1$ we prove that these maximal operators do not increase the $L^p$-variation of a function for any $p \geq 1$, while in dimensions $d>1$ we obtain the corresponding results for the $L^2$-variation. Similar results are proved for the discrete versions of these operators.
