International Journal of Computational and Applied Mathematics & Computer Science
E-ISSN: 2769-2477
Volume 2, 2022
The Generalization of Fourier-transform and the Peter-weyl Theorem
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Abstract: This article is devoted to the generalization of the Fourier transform and harmonic analysis on compact Hausdorff groups, we construct the Fourier-Stieltjes calculus, which is associated with the semigroups on the Hilbert space. We obtain that let $$ U_{1}: L^{1}(G) \rightarrow L (H,H)$$ be a nondegenerate unitary representation then there exists a unique representation $$ U: G \rightarrow U (H)$$ such that $$ U_{1}= U_{st}$$. Also, we establish that assume $$ U:G \rightarrow U(H)$$ is a unitary representation of the group $$G$$ and assume $$ U_{st}:L^{1}(G) \rightarrow L(H,H)$$ is a unitary representation of functional space $$ L^{1}(G)$$ then there is a mapping $$ Y:U\rightarrow U_{st}$$, which is a bijection.
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Pages: 57-64
DOI: 10.37394/232028.2022.2.11