**Rajendra G. Vyas**

## Fourier
Series with Small Gaps

**Abstract:**

Let $f$ be a $2\pi$-periodic function in $L^1[-\pi,\pi]$ and $\sum\limits_{k=-\infty}^\infty
\widehat{f}(n_k) e^{in_kx}$ be its lacunary Fourier series with small gaps. We
have estimated Fourier coefficients of $f$ if it is of $\varphi \bigwedge BV$
locally. We have also obtained a precise interconnection between the lacunarity
in such series and the localness of the hypothesis to be satisfied by the
generic function which allows us to the interpolate the results concerning
lacunary series and non-lacunary series.

**Keywords:**

Fourier series with small gaps, Order of magnitude of Fourier coefficients,
$\Phi-\bigwedge$-bounded variations.

**MSC 2000:** 42A16, 42A55