E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. Volume II: Structure and Analysis for Compact Groups. Analysis on for Compact Groups. Analysis on Locally Compact Abelian Groups. (D. Grundl. d. math. Wiss., Bd. 152). IX + 771 S. Berlin/Heidelberg/ New York 1970. Springer - Verlag. Preis geb. DM 140, –

1971 ◽  
Vol 51 (3) ◽  
pp. 238-238
Author(s):  
M. Landsberg
Keyword(s):  
New York ◽  
Harmonic Analysis ◽  
Abelian Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups

scholarly journals Locally compact products and coproducts in categories of topological groups

1977 ◽  
Vol 17 (3) ◽  
pp. 401-417 ◽  
Author(s):  
Karl Heinrich Hofmann ◽  
Sidney A. Morris
Keyword(s):  
Abelian Groups ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Topological Groups ◽  
Locally Compact ◽  
Torsion Free Group ◽  
Locally Compact Abelian Groups ◽  
Discrete Torsion ◽  
Compact Abelian Groups ◽  
Torsion Free

In the category of locally compact groups not all families of groups have a product. Precisely which families do have a product and a description of the product is a corollary of the main theorem proved here. In the category of locally compact abelian groups a family {Gj; j ∈ J} has a product if and only if all but a finite number of the Gj are of the form Kj × Dj, where Kj is a compact group and Dj is a discrete torsion free group. Dualizing identifies the families having coproducts in the category of locally compact abelian groups and so answers a question of Z. Semadeni.

scholarly journals Generalized hypergroups and orthogonal polynomials

1996 ◽  
Vol 142 ◽  
pp. 67-93 ◽  
Author(s):  
Nobuaki Obata ◽  
Norman J. Wildberger
Keyword(s):  
Orthogonal Polynomials ◽  
Harmonic Analysis ◽  
Abelian Groups ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Convolution Algebras ◽  
Compact Abelian Groups ◽  
Compact Hypergroup

We study in this paper a generalization of the notion of a discrete hypergroup with particular emphasis on the relation with systems of orthogonal polynomials. The concept of a locally compact hypergroup was introduced by Dunkl [8], Jewett [12] and Spector [25]. It generalizes convolution algebras of measures associated to groups as well as linearization formulae of classical families of orthogonal polynomials, and many results of harmonic analysis on locally compact abelian groups can be carried over to the case of commutative hypergroups; see Heyer [11], Litvinov [17], Ross [22], and references cited therein. Orthogonal polynomials have been studied in terms of hypergroups by Lasser [15] and Voit [31], see also the works of Connett and Schwartz [6] and Schwartz [23] where a similar spirit is observed.

A MACWILLIAMS FORMULA FOR CONVOLUTIONAL CODES

2007 ◽  
Vol 03 (02) ◽  
pp. 191-206
Author(s):  
PATRICK SOLÉ ◽  
DMITRII ZINOVIEV
Keyword(s):  
Functional Equation ◽  
Zeta Function ◽  
Harmonic Analysis ◽  
Abelian Groups ◽  
Convolutional Codes ◽  
Weight Enumerator ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups

Regarding convolutional codes as polynomial analogues of arithmetic lattices, we derive a Poisson–Jacobi formula for their trivariate weight enumerator. The proof is based on harmonic analysis on locally compact abelian groups as developed in Tate's thesis to derive the functional equation of the zeta function.

Orderings in locally compact Abelian groups and the theorem of F. and M. Riesz

1983 ◽  
Vol 93 (3) ◽  
pp. 441-457 ◽  
Author(s):  
Edwin Hewitt ◽  
Shozo Koshi
Keyword(s):  
Harmonic Analysis ◽  
Abelian Groups ◽  
Complete Classification ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups

Background (1·1). Ordered Abelian groups have been studied for nearly a century. Since the early 1950's, it has been recognized that orderings in locally compact Abelian groups can play an important rôle in harmonic analysis on such groups. In this paper we study orderings, especially in topological Abelian groups with either topological or measure-theoretic properties, obtaining nearly a complete classification of such orderings. We then apply these results to determine the limitations of the celebrated theorem of F. and M. Riesz on such groups.

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