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I believe the Sugiura article is, from MathSciNet, (MR0294571 Reviewed) Sugiura, Mitsuo Fourier series of smooth functions on compact Lie groups. Osaka J. Math. 8 (1971), 33–47. Apart from deciding on a summation order, there is the more pointed issue of an upper bound for sup-norm in terms of $L^2$ norm of functions in the irreducibles, obviously far ...

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