Small gaps Fourier series and generalized variations

Suppose f is an element of L [-pi , pi] has a Fourier series Sigma (infinity)(k) = -infinity (f) over cap (n(k) ) ei(nkx) (n(-k) = -n(k)) with small gaps n(k+1) - n(k) >= q 1 for all k >= 0. Here, by applying the Wiener-Ingham result for finite trigonometric sum with 'small' gaps, we estimate the order of magnitude of tPhe Fourier coefficients and obtain a sufficient condition for the convergence of the series Sigma(k is an element of z) vertical bar (f) over cap (n(k)) vertical bar(beta) (0 < beta <= 2 ) if f is locally of class Lambda BV (P (n)) up arrow infinity, phi).