Functions whose Fourier series converge for every change of variable

Abstract

1. A theorem of Pal and Bohr, [l], [2], asserts that for every continuous/ of period 27r there is a homeomorphism g of [ — ir, ir] with itself such that the Fourier series of/og converges uniformly. Salem [3] has given a powerful test for the uniform convergence of a Fourier series. On the other hand, there is no criterion which gives necessary and sufficient conditions for the Fourier series of a continuous function to converge everywhere. In this note we show that the method of Salem may be used to determine a necessary and sufficient condition that a continuous function / be such that the Fourier series of / o g should converge everywhere for every homeomorphism g of [ — ir, ir] with itself. It is clear that this condition must be weaker than bounded variation since the continuous functions of bounded ^-variation with 4> = ^, p>l, have uniformly convergent Fourier series, and this class of functions is preserved by composition with homeomorphisms [3], [4].