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Distributional point values and convergence of Fourier series and integrals. (English) Zbl 1138.46030
The authors show that the distributional point values of a tempered distribution in the sense of Łojasiewicz are characterized by their Fourier transforms. More precisely, for a tempered distribution $f$ whose Fourier transform is locally integrable, the distribution has the point value $\gamma$ at some point ${x}_{0}$ if and only if the integral for the Fourier transform of $\stackrel{^}{f}$ converges to $\gamma$ in the Cesàro sense. This result generalizes an earlier result of the second author on the characterization of the Fourier series at points where the the distributional point value exists. They also apply their results to lacunary series of distributions.
##### MSC:
 46F05 Topological linear spaces of test functions, distributions and ultradistributions 42A55 Lacunary series of trigonometric and other functions; Riesz products 42B10 Fourier type transforms, several variables