Harmonic Structures of Hilbert Spaces over Compact Abelian Groups
Keywords:
Compact abelian groups, Harmonic analysis, Reproducing kernel Hilbert spaces, Banach algebras, Fourier analysisAbstract
This paper investigates harmonic Hilbert spaces defined over compact abelian groups from a functional analytic and harmonic analysis perspective. These spaces are constructed via Fourier analytic techniques on the dual group, incorporating suitable weighting functions to control regularity and algebraic behavior. We examine conditions under which such Hilbert spaces admit a reproducing kernel structure and remain stable under pointwise multiplication of functions.
Special attention is given to identifying sufficient assumptions on the associated weight functions that guarantee the resulting spaces form Banach algebras equipped with a natural involution induced by complex conjugation. Under these assumptions, the studied spaces exhibit symmetry properties that allow them to be treated as invaolutive Banach algebras within a harmonic framework. Spectral characteristics of these algebras are also analyzed, and criteria are provided ensuring that their spectra coincide with those of the algebra of continuous functions on the underlying compact group.
Furthermore, the paper explores parametric families of harmonic Hilbert spaces generated by semigroups of self-adjoint operators acting on . Within this setting, a close relationship is established between convolution-type bounds on the defining functions and the existence of a compatible algebraic structure. Finally, connections between the proposed harmonic Hilbert spaces and certain Fourier-type function spaces are discussed, highlighting their relevance to approximation theory and harmonic analysis on compact groups.
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