On the uniqueness problem for Fourier series (Q1769728)

Denote by \(S_n(x, f)\), \(n= 1,2,\dots\), the partial sums of the Fourier series of a function \(f\in L_1(-\pi,\pi)\). The following theorem is proved: Let \(0<\alpha<1\) be such that \[ \int^\pi_{-\pi} \int^\pi_0 t^{-1-\alpha}|f(x+ t)- f(x)|\,dx\,dt< \infty, \] and let \(E\) be a subset of \((-\pi,\pi)\) such that \[ \sum^\infty_{n=1} 2^{n(1-\alpha)} C_{1-\alpha}(E_n(x_0))= \infty \] for almost all points \(x_0\in (-\pi,\pi)\), where \[ E_n(x_0):= \{x\not\in E: 2^{-1-n}<|x- x_0|< 2^{-n}\}. \] Under these conditions, if \[ \lim_{n\to\infty}\, S_n(x,f)= 0\quad\text{at each point }x\not\in E, \] then \(f(x)= 0\) almost everywhere on \((-\pi,\pi)\). We recall that the \(\sigma\)-capacity of a Borel set \(E\) is defined by \[ C_\sigma(E):= \Biggl(\inf_{\mu\prec E}\,\int_E \int_E {d\mu(x)d\mu(y)\over|x- y|^\sigma}\Biggr)^{-1}, \] where \(\mu\prec E\) means that \(d\mu\) is a probability measure with support in \(E\).