Related papers: Harmonic Synthesis on Group Extensions
In his classical paper, Laurent Schwartz proved that on the real line, in every linear translation invariant space of continuous complex valued functions, which is closed under compact convergence the exponential monomials span a dense…
In this paper we solve a long-standing problem which goes back to Laurent Schwartz's work on mean periodic functions. Namely, we completely characterise those locally compact Abelian groups having spectral synthesis. So far a…
In a former paper we introduced the concept of localisation of ideals in the Fourier algebra of a locally compact Abelian group. It turns out that localisability of a closed ideal in the Fourier algebra is equivalent to the synthesisability…
A compactly generated group is noncompact if and only if it admits a nonconstant harmonic function (for some, equivalently for every, reasonable measure). This generalizes the known fact that a finitely generated group is infinite if and…
Recently we introduced the concept of localisability of ideals in the Fourier algebra of locally compact Abelian groups. It turns out that localisability can be used to characterise synthesisability of varieties. Based on this we show that…
The interplay between the invariant subspace theory and spectral synthesis for locally compact abelian group discovered by Arveson is extended to include other topics as harmonic analysis for Varopoulos algebras and approximation by…
We extend a theorem by Kleiner, stating that on a group with polynomial growth, the space of harmonic functions of polynomial of at most $k$ is finite dimensional, to the settings of locally compact groups equipped with measures with…
In our former paper we introduced the concept of localisation of ideals in the Fourier algebra of a locally compact Abelian group. It turns out that localisability of a closed ideal in the Fourier algebra is equivalent to the…
We study the connection between the dimension of certain spaces of harmonic functions on a group and its geometric and algebraic properties. Our main result shows that (for sufficiently "nice" random walk measures) a connected, compactly…
It is a simple fact that a subgroup generated by a subset $A$ of an abelian group is the direct sum of the cyclic groups $\langle a\rangle$, $a\in A$ if and only if the set $A$ is independent. In [5] the concept of an $independent$ set in…
Over a $p$-adic local field $F$ of characteristic zero, we develop a new type of harmonic analysis on an extended symplectic group $G={\mathbb G}_m\times{\mathrm Sp}_{2n}$. It is associated to the Langlands $\gamma$-functions attached to…
We prove various results connecting structural or algebraic properties of graphs and groups to conditions on their spaces of harmonic functions. In particular: we show that a group with a finitely supported symmetric measure has a…
This paper studies certain aspects of harmonic analysis on nonabelian free groups. We focus on the concept of a positive definite function on the free group and our primary goal is to understand how such functions can be extended from balls…
In this paper we show that spectral analysis implies spectral synthesis for arbitrary varieties on locally compact Abelian groups, which have no discrete subgroup of infinite torsion free rank..
A differential form defined on a Riemannian manifold is said to harmonic if it is closed and co-closed. Harmonic differential forms are a natural multi-dimensional extension of the concept of analytic function of complex variable. In this…
This paper presents theoretical analysis and software implementation for real harmonics analysis on the special orthogonal group. Noncommutative harmonic analysis for complex-valued functions on the special orthogonal group has been studied…
The period is a classical complex analytic invariant for a compact Riemann surface defined by integration of differential 1-forms. It has a strong relationship with the complex structure of the surface. In this chapter, we review another…
In this work we study the structure of finitely generated groups for which a space of harmonic functions with fixed polynomial growth is finite dimensional. It is conjectured that such groups must be virtually nilpotent (the converse…
The family of harmonic Hilbert spaces is a natural enlargement of those classical $L^{2}$-Sobolev space on $\R^d$ which consist of continuous functions. In the present paper we demonstrate that the use of basic results from the theory of…
We define and study entanglement of continuous positive definite functions on products of compact groups. We formulate and prove an infinite-dimensional analog of Horodecki Theorem, giving a necessary and sufficient criterion for…