Multipliers from L 1 (G) to a reflexive segal algebra

1991 ◽  
pp. 163-168
Author(s):  
Guangzhong Ouyang
Keyword(s):  
Segal Algebra

scholarly journals A Sequential Approach to Mild Distributions

Axioms ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 25 ◽  
Author(s):  
Hans G. Feichtinger
Keyword(s):  
Fourier Transform ◽  
Number System ◽  
Continuous Functions ◽  
Sequential Approach ◽  
Tempered Distributions ◽  
Segal Algebra ◽  
Gabor Analysis ◽  
Plancherel’S Theorem ◽  
Lca Groups ◽  
Gelfand Triple

The Banach Gelfand Triple ( S 0 , L 2 , S 0 ′ ) ( R d ) consists of S 0 ( R d ) , ∥ · ∥ S 0 , a very specific Segal algebra as algebra of test functions, the Hilbert space L 2 ( R d ) , ∥ · ∥ 2 and the dual space S 0 ′ ( R d ) , whose elements are also called “mild distributions”. Together they provide a universal tool for Fourier Analysis in its many manifestations. It is indispensable for a proper formulation of Gabor Analysis, but also useful for a distributional description of the classical (generalized) Fourier transform (with Plancherel’s Theorem and the Fourier Inversion Theorem as core statements) or the foundations of Abstract Harmonic Analysis, as it is not difficult to formulate this theory in the context of locally compact Abelian (LCA) groups. A new approach presented recently allows to introduce S 0 ( R d ) , ∥ · ∥ S 0 and hence ( S 0 ′ ( R d ) , ∥ · ∥ S 0 ′ ) , the space of “mild distributions”, without the use of the Lebesgue integral or the theory of tempered distributions. The present notes will describe an alternative, even more elementary approach to the same objects, based on the idea of completion (in an appropriate sense). By drawing the analogy to the real number system, viewed as infinite decimals, we hope that this approach is also more interesting for engineers. Of course it is very much inspired by the Lighthill approach to the theory of tempered distributions. The main topic of this article is thus an outline of the sequential approach in this concrete setting and the clarification of the fact that it is just another way of describing the Banach Gelfand Triple. The objects of the extended domain for the Short-Time Fourier Transform are (equivalence classes) of so-called mild Cauchy sequences (in short ECmiCS). Representatives are sequences of bounded, continuous functions, which correspond in a natural way to mild distributions as introduced in earlier papers via duality theory. Our key result shows how standard functional analytic arguments combined with concrete properties of the Segal algebra S 0 ( R d ) , ∥ · ∥ S 0 can be used to establish this natural identification.

scholarly journals ESSENTIAL AMENABILITY OF ABSTRACT SEGAL ALGEBRAS

2009 ◽  
Vol 79 (2) ◽  
pp. 319-325 ◽  
Author(s):  
H. SAMEA
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Segal Algebra ◽  
Amenable Banach Algebra ◽  
Segal Algebras

AbstractA number of well-known results of Ghahramani and Loy on the essential amenability of Banach algebras are generalized. It is proved that a symmetric abstract Segal algebra with respect to an amenable Banach algebra is essentially amenable. Applications to locally compact groups are given.

scholarly journals APPROXIMATE AMENABILITY OF SEGAL ALGEBRAS

2013 ◽  
Vol 95 (1) ◽  
pp. 20-35 ◽  
Author(s):  
MAHMOOD ALAGHMANDAN
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Amenable Group ◽  
Locally Compact Abelian Group ◽  
Compact Groups ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Segal Algebra ◽  
Approximate Amenability ◽  
Segal Algebras

AbstractIn this paper we first show that for a locally compact amenable group $G$, every proper abstract Segal algebra of the Fourier algebra on $G$ is not approximately amenable; consequently, every proper Segal algebra on a locally compact abelian group is not approximately amenable. Then using the hypergroup generated by the dual of a compact group, it is shown that all proper Segal algebras of a class of compact groups including the $2\times 2$ special unitary group, $\mathrm{SU} (2)$, are not approximately amenable.

scholarly journals Constructions of the maximal strongly character invariant segal algebras and their applications

1983 ◽  
Vol 35 (1) ◽  
pp. 123-131
Author(s):  
Chang-Pao Chen
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Banach Algebras ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Segal Algebra ◽  
Segal Algebras

AbstractLet G denote any locally compact abelian group with the dual group Γ. We construct a new kind of subalgebra L1(G) ⊗ΓS of L1(G) from given Banach ideal S of L1(G). We show that L1(G) ⊗гS is the larger amoung all strongly character invariant homogeneous Banach algebras in S. when S contains a strongly character invariant Segal algebra on G, it is show that L1(G) ⊗гS is also the largest among all strongly character invariant Segal algebras in S. We give applications to characterizations of two kinds of subalgebras of L1(G)-strongly character invariant Segal algebras on G and Banach ideal in L1(G) which contain a strongly character invariant Segal algebra on G.

scholarly journals A note on multipliers on a Segal algebra

1974 ◽  
Vol 49 (2) ◽  
pp. 125-127 ◽  
Author(s):  
K. Unni
Keyword(s):  
Segal Algebra

Weak amenability of certain classes of Banach algebras without bounded approximate identities

2002 ◽  
Vol 133 (2) ◽  
pp. 357-371 ◽  
Author(s):  
F. GHAHRAMANI ◽  
A. T. M. LAU
Keyword(s):  
Banach Algebras ◽  
Locally Compact Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Segal Algebra ◽  
Weak Amenability ◽  
Approximate Identities ◽  
Dual Module ◽  
Arens Regularity ◽  
Segal Algebras

In a recent paper [3] Dales and Pandey have shown that the class Sp of Segal algebras is weakly amenable. In this paper, for various classes of Segal algebras, we characterize derivations and multipliers from a Segal algebra into itself and into its dual module. In particular, we prove that every Segal algebra on a locally compact abelian group is weakly amenable and an abstract Segal subalgebra of a commutative weakly amenable Banach algebra is weakly amenable. We also introduce the Lebesgue–Fourier algebra of a locally compact group G and study its Arens regularity when G is discrete or compact.

scholarly journals Every Segal algebra satisfies Ditkin's condition

1971 ◽  
Vol 40 (3) ◽  
pp. 235-237 ◽  
Author(s):  
Leonard Yap
Keyword(s):  
Segal Algebra
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