Characterization of the Fourier series of a distribution having a value at a point

Abstract:

Let $f$ be a periodic distribution of period $2\pi$. Let $\sum ^\infty _{n=-\infty } a_ne^{in\theta }$ be its Fourier series. We show that the distributional point value $f(\theta _0)$ exists and equals $\gamma$ if and only if the partial sums $\sum _{-x\le n\le ax}a_ne^{in\theta _0}$ converge to $\gamma$ in the Cesàro sense as $x\to \infty$ for each $a>0$.