Wiener–Ingham type inequality for Vilenkin groups and its application to harmonic analysis

Abstract

An important trigonometric inequality essentially due to Wiener but later on made precise by Ingham concerning the lacunary trigonometric sums $f(x)=\sum A_ke^{in_kx}$, where $A_k$’s are complex numbers, $n_{-k}=-n_k$ and $\{n_k\}$ satisfies the small gap condition $(n_{k+1}-n_k)\ge q\ge 1$ for $k=0,1,2,\ldots$, says that if I is any subinterval of $[-\pi ,\pi ]$ of length $|I|=2\pi (1+\delta )/q>2\pi /q$ then $\sum |A_k|^2\le A_{\delta }|I|^{-1}\int _I|f|^2$, $|A_k|\le A_{\delta }|I|^{-1}\int _I|f|$, wherein $A_{\delta }$ depends only on $\delta$. Such an inequality is proved here in the setting of the Vilenkin groups G. The inequality is then applied to generalize the Bernstěin, Szász and Stečhkin type results concerning the absolute convergence of Fourier series on G.