Divergent Fourier series: Numerical experiments (Q808363)

The divergence rate in Fourier series as given by \textit{J.-P. Kahane} [Some random series of functions (1968; Zbl 0192.538)] in the form \[ \limsup_{n\to \infty}(| S_ n(f,x)| /A_ n)=\infty,\quad A_ n=o(\log \log n) \] is discussed \((S_ n(f,x)\) are partial sums). A new point of view on this example, a probabilistic interpretation of the Kahane example, is proposed. The probabilistic nature of the \(S_ n\) sums behaves essentially as a typical sequence of record values associated to a sequence of independent identically distributed random variables with double exponential tail. The behavior of the sequence of record values \(\tilde S_ n\) given by the formula \(\tilde S_ n\equiv \sum_{i=5}m_ i(\sin n\theta_ i/\theta_ i)\) is checked, numerical values are calculated, the results are tabulated and plotted. The results are valid only for the Kahane type examples.