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Rearrangement of coefficients of Fourier series on \(SU_ 2\). (English) Zbl 0621.43011
It is classical that if a Fourier series f (on the circle) remains in \(L^ 1\) under all permutations of the coefficients, then \(f\in L^ 2\). This was extended by S. Helgason [Proc. Am. Math. Soc. 9, 782-790 (1958; Zbl 0091.109)] to compact abelian groups. The author studies the group \(SU_ 2\) and shows that, for central functions, with \(L^ p\) and \(L^ q\) replacing \(L^ 1\) and \(L^ 2\), the theorem can be extended to \(3/2<p<q\leq 2\), fails for \(p<3/2\leq q\) and for \(3/2<p\leq 2<q\), and is meaningless for \(p\geq 3\) (only trigonometric polynomials remain in \(L^ 3\) for every permutation). For arbitrary \(L^ p(SU_ 2)\)-functions a natural generalization fails for \(p<2\), \(q\geq 2\).
Reviewer: R.P.Boas
43A75 Harmonic analysis on specific compact groups
43A77 Harmonic analysis on general compact groups
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