Peter–Weyl Theorem
A foundational theorem in harmonic analysis on compact groups, generalizing classical Fourier analysis to non-commutative compact groups. It states that the space of square-integrable functions on a compact group decomposes orthogonally into a direct sum of subspaces spanned by matrix coefficients of all irreducible unitary representations.
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The Peter–Weyl theorem decomposes the space L^2(G) of square-integrable functions on a compact group into an orthogonal direct sum of subspaces spanned by the matrix coefficients of all irreducible unitary representations, with each irreducible representation appearing with a multiplicity equal to its dimension.
The Sobolev space H^s can be equivalently described as the space of functions whose Peter–Weyl expansion coefficients remain square-integrable after being weighted by the s-th power of the eigenvalues; the larger s is, the smoother the function and the faster the high-frequency coefficients decay.
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The Peter–Weyl theorem decomposes the space L^2(G) of square-integrable functions on a compact group into an orthogonal direct sum of subspaces spanned by the matrix coefficients of all irreducible unitary representations, with each irreducible representation appearing with a multiplicity equal to its dimension.
50%UnverifiedThe Sobolev space H^s can be equivalently described as the space of functions whose Peter–Weyl expansion coefficients remain square-integrable after being weighted by the s-th power of the eigenvalues; the larger s is, the smoother the function and the faster the high-frequency coefficients decay.
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