Limits of indeterminacy of trigonometric series
The author proves the following main theorem: for any trigonometric series: \(\sum^{\infty}_{n=0}c_ n \cos (nx+\theta_ n)\), let \(S_ n(x)=\sum^{n}_{m=0}c_ m \cos (mx+\theta_ m)\) and \(\bar S(x)= \limsup_{n\to \infty}S_ n(x)=F(x)\), \(\bar S(x)= \limsup_{n\to \infty}S_ n(x)=g(x)\), then \(mes\{x\in [-\pi,\pi]:\) \(-\infty <\underline S(x)\leq S(x)=+\infty \}=0.\) In particular \(mes\{x\in [-\pi,\pi]:\) \(\lim S(x)=\infty \}=0.\) This theorem is closely related to the work of \textit{D. E. Menchoff} [C. R. Acad. Sci. URSS, Nov. Ser. 26, 214-216 (1940; Zbl 0023.02901)]. By the end of this paper the author poses four interesting problems related to the above theorem.
scientific article; zbMATH DE number 4127837