## 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 $$E$$.

### MSC:

 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

### Keywords:

partial sums; Fourier series; uniqueness; capacity; potential theory
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