A Covariance Equation

• Published in: The Journal of geometric analysis 2020-12, Vol.30 (4), p.3398-3412
• Main Author: Youssfi, El Hassan
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Continuity (mathematics); Convex and Discrete Geometry; Covariance; Differential Geometry; Dynamical Systems and Ergodic Theory; Fourier Analysis; Geometry; Global Analysis and Analysis on Manifolds; Group theory; Mathematics; Mathematics and Statistics; Topology
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-019-00201-7

Let S be a commutative semigroup with identity e and let Γ be a compact subset in the pointwise convergence topology of the space S ′ of all non-zero multiplicative functions on S . Given a continuous function F : Γ → C and a complex regular Borel measure μ on Γ such that μ ( Γ ) ≠ 0 . It is shown that μ ( Γ ) ∫ Γ ϱ ( s ) ϱ ( t ) ¯ | F | 2 ( ϱ ) d μ ( ϱ ) = ∫ Γ ϱ ( s ) F ( ϱ ) d μ ( ϱ ) ∫ Γ ϱ ( t ) F ( ϱ ) ¯ d μ ( ϱ ) for all ( s , t ) ∈ S × S if and only if for some γ ∈ Γ , the support of μ is contained in { F = 0 } ∪ { γ } . Several applications of this characterization are derived. In particular, the reduction of our theorem to the semigroup of non-negative integers ( N 0 , + ) solves a problem posed by El Fallah, Klaja, Kellay, Mashregui, and Ransford in a more general context. More consequences of our results are given, some of them illustrate the probabilistic flavor behind the problem studied herein and others establish extremal properties of analytic kernels.