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Functional/Harmonic Analysis of Covariant Transforms

Periodic Reporting for period 1 - covtrans (Functional/Harmonic Analysis of Covariant Transforms)

Reporting period: 2019-03-01 to 2021-02-28

The covariant transforms/functions of characters (one-dimensional continuous irreducible unitary representations) of closed subgroups have applications in different mathematical areas such as number theory (automorphic forms), induced representations, homogeneous spaces, complex (hypercomplex) analysis, and coherent states. In general, classical harmonic analysis methods cannot be employed as a unified theory for covariant functions of a given closed subgroup. In this project, we developed harmonic/functional analysis foundations for covariant functions of characters.

We developed some operator theoretic aspects related to Banach covariant function spaces of characters of normal or compact subgroups including a unified theory for structure of Banach convolution modules induced by the group algebras on Banach covariant function spaces of characters of subgroups (normal or compact) and covariant convolutions (convolution of covariant functions).

The introduced structures and properties of covariant function spaces imply a better understanding of convolution type covariant transforms and presents a unified theory for harmonic/functional analysis of covariant functions/transforms. The presented theory can be applied in different directions including mathematical physics (coherent states), representation theory of groups, postmodern harmonic analysis. It also generalizes classical methods of abstract harmonic analysis on quotient groups and homogeneous spaces of compact subgroups.
In the direction of academic development plans, the fellow attended many training courses, workshop, and conferences.

In the direction of research, the fellow worked on development of the mathematical theory of covariant functions from different perspectives. A website is created which informed about project’s targets and achieved progress. In line with the policy for dissemination of our results, we make our papers online on arXiv once they submitted to journals for review process.

For compact subgroups, the developed harmonic analysis methods for covariant functions of characters based on an operator theoretic approach. The results include interesting characterizations for classical Banach spaces on covariant functions of characters of compact subgroups. We classified classical Banach spaces of covariant functions of characters of compact subgroups as quotient spaces of classical Banach spaces of functions on the group. These classifications imply a unified and constructive characterizations for the dual spaces of classical Banach spaces of covariant funcions of characters of compact subgroups. We also investigated different algebraic and analytic properties of these spaces including module properties and convolutions.

In the case of normal subgroups, we developed a unified operator theoretic approach related to Banach covariant function spaces of characters of normal subgroups including a theory for structure of Banach convolution modules induced by the group algebras on Banach covariant function spaces of characters of normal subgroups and covariant convolutions (convolution of covariant functions). For general characters, we studied properties of convolution module actions induced by group algebras on classical Banach spaces of covariant functions. In the case that the character is invariant, we introduced a well-defined notion of convolution for covariant functions.
The introduced characterizations and investigated structures/properties of covariant function spaces of characters can be employed as a unified theory for harmonic/functional analysis methods of covariant functions of characters. In addition, the presented theory imply better understanding of convolution type covariant transforms. It will substantially contribute to variant fields such as algebra, analysis and theoretical physics.
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