Wiener-Ingham type inequality for Vilenkin groups and its application to harmonic analysis (Q520015)

An important trigonometric inequality essentially due to Wiener but later on made precise by Ingham concerning the lacunary trigonometric sums \(f(x) = \sum A_k e^{in_k x}\), where the \(A_k\)'s are complex numbers, \(n_{-k} = -n_k\) and \(\{n_k \}\) satisfies the small gap condition \((n_{k+1} - n_k) \geq q \geq 1\) 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 \(\sum | A_k | ^2 \leq A_{\delta} | I| ^{-1}\int_{I} | f | ^2, | A_k | \leq A_{\delta} | I| ^{-1}\int_{I} | f | \), wherein \(A_{\delta}\) depends only on \(\delta\). In their paper the authors prove such an inequality 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 \(G\).