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
whose Fourier transform is locally integrable, the distribution has the point value
at some point
if and only if the integral for the Fourier transform of
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.