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An extension of deLeeuw’s theorem to the n -dimensional rotation group

Anthony H. Dooley, Garth I. Gaudry (1984)

Annales de l'institut Fourier

We study a method of approximating representations of the group M ( n ) by those of the group S O ( n + 1 ) . As a consequence we establish a version of a theorem of DeLeeuw for Fourier multipliers of L p that applies to the “restrictions” of a function on the dual of M ( n ) to the dual of S O ( n + 1 ) .

Convolutions on compact groups and Fourier algebras of coset spaces

Brian E. Forrest, Ebrahim Samei, Nico Spronk (2010)

Studia Mathematica

We study two related questions. (1) For a compact group G, what are the ranges of the convolution maps on A(G × G) given for u,v in A(G) by u × v ↦ u*v̌ (v̌(s) = v(s^-1)) and u × v ↦ u*v? (2) For a locally compact group G and a compact subgroup K, what are the amenability properties of the Fourier algebra of the coset space A(G/K)? The algebra A(G/K) was defined and studied by the first named author. In answering the first question, we obtain, for compact groups which do not...

Haar system on a product of zero-dimensional compact groups

Sergei Lukomskii (2011)

Open Mathematics

In this work, we study the problem of constructing Haar bases on a product of arbitrary compact zero-dimensional Abelian groups. A general scheme for the construction of Haar functions is given for arbitrary dimension. For dimension d=2, we describe all Haar functions.

Measures and lacunary sets

Pascal Lefèvre (1999)

Studia Mathematica

We establish new connections between some classes of lacunary sets. The main tool is the use of (p,q)-summing or weakly compact operators (for Riesz sets). This point of view provides new properties of stationary sets and allows us to generalize to more general abelian groups than the torus some properties of p-Sidon sets. We also construct some new classes of Riesz sets.

Shift-modulation invariant spaces on LCA groups

Carlos Cabrelli, Victoria Paternostro (2012)

Studia Mathematica

A (K,Λ) shift-modulation invariant space is a subspace of L²(G) that is invariant under translations along elements in K and modulations by elements in Λ. Here G is a locally compact abelian group, and K and Λ are closed subgroups of G and the dual group Ĝ, respectively. We provide a characterization of shift-modulation invariant spaces when K and Λ are uniform lattices. This extends previous results known for L ² ( d ) . We develop fiberization techniques and suitable range functions adapted to LCA groups...

Some remarks on quasi-Cohen sets

Pascal Lefèvre, Daniel Li (2001)

Colloquium Mathematicae

We are interested in Banach space geometry characterizations of quasi-Cohen sets. For example, it turns out that they are exactly the subsets E of the dual of an abelian compact group G such that the canonical injection C ( G ) / C E c ( G ) L ² E ( G ) is a 2-summing operator. This easily yields an extension of a result due to S. Kwapień and A. Pełczyński. We also investigate some properties of translation invariant quotients of L¹ which are isomorphic to subspaces of L¹.

Spectral synthesis and operator synthesis

K. Parthasarathy, R. Prakash (2006)

Studia Mathematica

Relations between spectral synthesis in the Fourier algebra A(G) of a compact group G and the concept of operator synthesis due to Arveson have been studied in the literature. For an A(G)-submodule X of VN(G), X-synthesis in A(G) has been introduced by E. Kaniuth and A. Lau and studied recently by the present authors. To any such X we associate a V ( G ) -submodule X̂ of ℬ(L²(G)) (where V ( G ) is the weak-* Haagerup tensor product L ( G ) w * h L ( G ) ), define the concept of X̂-operator synthesis and prove that a closed set E...

Topological Dichotomy and Unconditional Convergence

Lefevre, Pascal (1999)

Serdica Mathematical Journal

In this paper, we give a criterion for unconditional convergence with respect to some summability methods, dealing with the topological size of the set of choices of sign providing convergence. We obtain similar results for boundedness. In particular, quasi-sure unconditional convergence implies unconditional convergence.

Uniformly cyclic vectors

Joseph Rosenblatt (2006)

Colloquium Mathematicae

A group acting on a measure space (X,β,λ) may or may not admit a cyclic vector in L ( X ) . This can occur when the acting group is as big as the group of all measure-preserving transformations. But it does not occur, even though there is no cardinality obstruction to it, for the regular action of a group on itself. The connection of cyclic vectors to the uniqueness of invariant means is also discussed.

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