In this article we show that the distributional point values of a tempered distribution are characterized by their Fourier transforms in the following way: If f ∈ S (R) and x 0 ∈ R, and f is locally integrable, then f (x 0 ) = γ distributionally if and only if there exists k such that 1 2π lim x→∞ ax −xf (t)e −ix 0 t dt = γ (C, k) , for each a > 0, and similarly in the case when f is a general distribution. Here (C, k) means in the Cesàro sense. This result generalizes the characterization of Fourier series of distributions with a distributional point value given in [5] by lim x→∞ −x≤n≤ax a n e inx 0 = γ (C, k) .We also show that under some extra conditions, as if the sequence {a n } ∞ n=−∞ belongs to the space l p for some p ∈ [1, ∞) and the tails satisfy the estimate ∞ |n|≥N |a n | p = O N 1−p , as N → ∞, the asymmetric partial sums converge to γ . We give convergence results in other cases and we also consider the convergence of the asymmetric partial integrals. We apply these results to lacunary Fourier series of distributions.