We show that two locally compact abelian groups Gx and G2 are isomorphic if there exists an algebra isomorphism T of L\Gλ) onto L\G2) with | | Γ | | < V2. This constant is best possible. The same result is proved for locally compact connected groups, but for the general locally compact group, the result is proved under the hypothesis | |T | |< 1.246. Similar results are given for the algebras C...

These notes are a gloss on the first chapter of Walter Rudin’s Fourier Analysis on Groups, and may be helpful to someone reading Rudin. The results I do prove are proved in more detail than they are in Rudin. I caution that before reading the first chapter of that book it is know about the Gelfand transform on commutative Banach algebras because results from that are used without even stating t...

In the case of a locally compact field K, the additive group (K,+) and the multiplicative group (K×, ·) play key roles in the theory. Each is a locally compact abelian group, hence amenable to the methods of Fourier analysis. Moreover, the additive group is self-Pontrjagin dual, and the multiplicative group K× is a target group for class field theory on K: that is, there is a bijective correspo...

We consider a class of functions, defined on a locally compact abelian group by letting a class of modulation operators act on a countable collection of functions. We derive sufficient conditions for such a class of functions to form a Bessel sequence or a frame and for two such systems to be dual frames. Explicit constructions are obtained via various generalizations of the classical B-splines...

‎Let $G$ be a locally compact abelian group‎. ‎The concept of a generalized multiresolution structure (GMS) in $L^2(G)$ is discussed which is a generalization of GMS in $L^2(mathbb{R})$‎. ‎Basically a GMS in $L^2(G)$ consists of an increasing sequence of closed subspaces of $L^2(G)$ and a pseudoframe of translation type at each level‎. ‎Also‎, ‎the construction of affine frames for $L^2(G)$ bas...

This paper develops several aspects of shift-invariant spaces on locally compact abelian groups. For a second countable locally compact abelian group G we prove a useful Hilbert space isomorphism, introduce range functions and give a characterization of shift-invariant subspaces of L 2 (G) in terms of range functions. Utilizing these functions, we generalize characterizations of frames and Ries...

A topological group $X$ is called $duoseparable$ if there exists a countable set $S\subseteq X$ such that $SUS=X$ for any neighborhood $U\subseteq of the unit. We construct functor $F$ assigning to each (abelian) duoseparable (abelain-by-cyclic) $FX$, containing an isomorphic copy $X$. In fact, defined on category unital topologized magmas. Also we prove $\sigma$-compact locally compact abelian...