Probability on Compact Lie Groups by David Applebaum

Author:David Applebaum
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Then since by ( 2.​3.​8),

and by uniqueness of Fourier coefficients (in the Hilbert space sense) for all .9

Since Haar measure is finite, , and so . Recall that by Theorem 2.2.4 , which is the algebra of all continuous functions on that have only finitely many non-zero Fourier coefficients, is norm dense in . Let . Then there exists a finite subset of so that

for all . Furthermore, by the Schur orthogonality relations, if . Using the Parseval-Plancherel identity ( 2.​3.​9), for each :

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