Orthogonal expansions related to compact Gelfand pairs

TY - JOUR

T1 - Orthogonal expansions related to compact Gelfand pairs

AU - Berg, Christian

AU - Peron, Ana P.

AU - Porcu, Emilio

PY - 2018

Y1 - 2018

N2 - For a locally compact group G, let P(G) denote the set of continuous positive definite functions [Formula presented]. Given a compact Gelfand pair [Formula presented] and a locally compact group L, we characterize the class [Formula presented] of functions f∈P(G×L) which are bi-invariant in the G-variable with respect to K. The functions of this class are the functions having a uniformly convergent expansion ∑ φ∈ZB(φ)(u)φ(x) for [Formula presented], where the sum is over the space Z of positive definite spherical functions [Formula presented] for the Gelfand pair, and (B(φ)) φ∈Z is a family of continuous positive definite functions on L such that ∑ φ∈ZB(φ)(e L)<∞. Here e L is the neutral element of the group L. For a compact Abelian group G considered as a Gelfand pair [Formula presented] with trivial K = {e G}, we obtain a characterization of P(G×L) in terms of Fourier expansions on the dual group [Formula presented]. The result is described in detail for the case of the Gelfand pairs [Formula presented] and [Formula presented] as well as for the product of these Gelfand pairs. The result generalizes recent theorems of Berg–Porcu (2016) and Guella–Menegatto (2016).

AB - For a locally compact group G, let P(G) denote the set of continuous positive definite functions [Formula presented]. Given a compact Gelfand pair [Formula presented] and a locally compact group L, we characterize the class [Formula presented] of functions f∈P(G×L) which are bi-invariant in the G-variable with respect to K. The functions of this class are the functions having a uniformly convergent expansion ∑ φ∈ZB(φ)(u)φ(x) for [Formula presented], where the sum is over the space Z of positive definite spherical functions [Formula presented] for the Gelfand pair, and (B(φ)) φ∈Z is a family of continuous positive definite functions on L such that ∑ φ∈ZB(φ)(e L)<∞. Here e L is the neutral element of the group L. For a compact Abelian group G considered as a Gelfand pair [Formula presented] with trivial K = {e G}, we obtain a characterization of P(G×L) in terms of Fourier expansions on the dual group [Formula presented]. The result is described in detail for the case of the Gelfand pairs [Formula presented] and [Formula presented] as well as for the product of these Gelfand pairs. The result generalizes recent theorems of Berg–Porcu (2016) and Guella–Menegatto (2016).

KW - Gelfand pairs

KW - Positive definite functions

KW - Primary

KW - Secondary

KW - Spherical functions

KW - Spherical harmonics for real an complex spheres

U2 - 10.1016/j.exmath.2017.10.005

DO - 10.1016/j.exmath.2017.10.005

M3 - Journal article

AN - SCOPUS:85033390130

SN - 0723-0869

VL - 36

SP - 259

EP - 277

JO - Expositiones Mathematicae

JF - Expositiones Mathematicae

IS - 3-4

ER -