Orthogonal expansions related to compact Gelfand pairs
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Orthogonal expansions related to compact Gelfand pairs. / Berg, Christian; Peron, Ana P.; Porcu, Emilio.
I: Expositiones Mathematicae, Bind 36, Nr. 3-4, 2018, s. 259-277.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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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
VL - 36
SP - 259
EP - 277
JO - Expositiones Mathematicae
JF - Expositiones Mathematicae
SN - 0723-0869
IS - 3-4
ER -
ID: 196170748