Problem Statement:Let G be a locally compact abelian group, and let H = 12(G). Consider the group of translationoperators {T}geG defined by Tg f(x) = f(x − g).1. Spectral Measure for Translation Operators: Construct the spectral measure E associated withthe unitary representation {T} and prove that it corresponds to the dual group G via theFourier transform.2. Measure-Theoretic Duality in Harmonic Analysis: Demonstrate that the Plancherel measure onGarises naturally from the spectral measure E, and prove that the Fourier transformintertwines the spectral measures of G and G.3. Spectral Decomposition and Harmonic Analysis Applications: Use the spectral measure E toperform a spectral decomposition of L2(G) and apply this decomposition to solve problems inharmonic analysis, such as convolution operators and translation-invariant subspaces.Requirements:••Apply spectral theory to unitary representations of locally compact groups.Utilize harmonic analysis and the Fourier transform in the construction of spectral measures.• Integrate measure-theoretic duality principles within the context of group representations.

Answered: Image /qna-images/answer/df676f20-96aa-42b5-93d0-87fcef647992.jpg

Answered: Problem Statement: Let G be a locally…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.6: Quotient Groups

Problem 11E: Find all homomorphic images of the quaternion group.

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