Let G be a locally compact hypergroup and let K sub-hypergroup of G. (G, K) is Gelfand pair if Mc(G//K), the algebra measures with support on double coset G//K, commutative for convolution. In this paper, assuming that pair, we define study Fourier transform then establish Plancherel theorem K).

In engineering and applied mathematics, Zak transforms have been effectively used for over 50 years in various applied settings. As Gelfand observed in a 1950 paper, the variable coefficient Fourier series ideas articulated in Andre Weil’s famous book on integration lead to an exceedingly elementary proof of the Plancherel Theorem for LCA groups. The transform for functions on R appearing in Za...

We introduce the concept of a modulator, which leads to a family of character formulae, each generalizing the Kirillov formula. For a suitable choice of modulator, this enables one to understand the Plancherel measure of a compact Lie group as arising from a partition of the identity on the dual of its Lie algebra.

We show that the Plancherel measure of the lamplighter random walk on a graph coincides with the expected spectral measure of the absorbing random walk on the Bernoulli percolation clusters. In the subcritical regime the spectrum is pure point and we construct a complete set of finitely supported eigenfunctions.

We study the principal series representations of central extensions of a split reductive algebraic group by a cyclic group of order n. We compute the Plancherel measure of the representation using Eisenstein series and a comparison method. In addition, we construct genuine central characters of the metaplectic torus in the simply-laced case.

Zero biasing for the normal approximation of a random variable W using Stein’s method was introduced in Goldstein and Reinert [16]. One instance in which the zero bias method may be applied is for W for which a Stein pair W,W ′ may be constructed, that is, for W that may be coupled to a variable W ′ such that W,W ′ is exchangeable and satisfies E(W ′|W ) = (1− a)W for some a ∈ (0, 1]. After giv...