On the point behavior of Fourier series and conjugate series (Q606473)

Summary: We investigate the point behavior of periodic functions and Schwartz distributions when the Fourier series and the conjugate series are both Abel summable at a point. In particular we show that if \(f\) is a bounded function and its Fourier series and conjugate series are Abel summable to values \(\gamma \) and \(\beta \) at the point \(\theta _{0}\), respectively, then the primitive of \(f\) is differentiable at \(\theta _{0}\), with derivative equal to \(\gamma \), the conjugate function satisfies \(\lim_{\theta \rightarrow \theta _0} \frac {3}{(\theta -\theta _0)^3} \int ^{\theta }_{\theta _0} \widetilde f(t)(\theta -t)^{2}\, dt = \beta \), and the Fourier series and the conjugate series are both \((C, \kappa )\) summable at \(\theta _{0}\), for any \(\kappa > 0\). We show a similar result for positive measures and \(L^{1}\) functions bounded from below. Since the converse of our results are valid, we therefore provide a complete characterization of simultaneous Abel summability of the Fourier and conjugate series in terms of ``average point values'', within the classes of positive measures and functions bounded from below. For general \(L^{1}\) functions, we also give a.e. distributional interpretation of \(-\frac 1{2\pi}\) p.v. \(\int ^{\pi }_{-\pi} f(t+ \theta _{0}) \cot \frac t2\,dt\) as the point value of the conjugate series when viewed as a distribution. We obtain more general results of this kind for arbitrary trigonometric series with coefficients of slow growth, i.e., periodic distributions.