Change of variable in Fourier expansions: Some old and new results (Q2782554)

Let \(f\) be a continuous function on the torus \(\mathbb{T}\), in symbol: \(f\in C(\mathbb{T})\). It is well known that the Fourier expansion \(\sum_{n\in\mathbb{Z}}\widehat f(n) e^{2\pi int}\) may diverge at some points, but one can remove the divergence by an appropriate change of variable. The author gives a concise survey of both the classical (nonstochastic) and random changes of variable and raises a number of open problems.NEWLINENEWLINENEWLINEThe idea to generate at random an increasing homeomorphism \(h\) of \([0,1]\) onto itself was devised by Ulam and Mauldin [see \textit{S. Graf}, \textit{R. D. Mauldin} and \textit{S. C. Williams}, Adv. Math. 60, 239-259 (1986; Zbl 0596.60005)]. Now the striking result proved by \textit{G. Kozma} and \textit{A. Olevskij} [Geom. Funct. Anal. 8, No. 6, 1016-1024 (1998; Zbl 0945.42003)] is the following:NEWLINENEWLINENEWLINE(i) For any \(f\in C(\mathbb{T})\), the Fourier partial sums of the superposition \(f\circ h\) satisfy almost surely the estimate \(\|S_N(f\circ h)\|= o(\log\log N)\).NEWLINENEWLINENEWLINE(ii) For any \(w(N)= o(\log\log N)\), there exists \(f\in C(\mathbb{T})\) such that with probability \(1\), \(\|S_N(f\circ h)\|> w(N)\) for infinitely many \(N\).NEWLINENEWLINEFor the entire collection see [Zbl 0980.00031].