On the uniqueness problem for Fourier series. (English) Zbl 1068.42007

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 \(E\).


42A63 Uniqueness of trigonometric expansions, uniqueness of Fourier expansions, Riemann theory, localization
31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions
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