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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
MSC:
 43A75 Harmonic analysis on specific compact groups 43A77 Harmonic analysis on general compact groups
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