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.

scholarly journals Isomorphisms of some convolution algebras and their multiplier algebras

1972 ◽  
Vol 7 (3) ◽  
pp. 321-335 ◽  
Author(s):  
U.B. Tewari ◽  
G.I. Gaudry
Keyword(s):  
Abelian Groups ◽  
Topological Groups ◽  
Locally Compact ◽  
Algebra Isomorphism ◽  
Locally Compact Abelian Groups ◽  
Convolution Algebras ◽  
Compact Abelian Groups ◽  
Multiplier Algebras

Let G1 and G2 be two locally compact abelian groups and let 1 ≤ p ∞. We prove that G1 and G2 are isomorphic as topological groups provid∈d there exists a bipositive or isometric algebra isomorphism of M(Ap (G1)) onto M(Ap (G2)). As a consequence of this, we prove that G1 and G2 are isomorphic as topological groups provided there exists a bipositive or isometric algebra isomorphism of Ap (G1) onto Ap (G2). Similar results about the algebras L1 ∩ Lp and L1 ∩ C0 are also established.

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.

Sign in / Sign up

Export Citation Format

Share Document