Approximation of distributions by convolutions in the Hausdorff metric

The paper deals with finding criteria for the Hausdorff convergence of sequences of convolution operators on quasi-Banach spaces of periodic realvalued distributions (generalized functions). In particular, the criteria for convergence on the Hardy classes, on the class of regular Borel measures, and on the class of pseudomeasures are found. These assertions are special cases of the general result obtained for rather wide collection of spaces. The given result relies essentially on the explicit description of the set of bounded convolution operators, acting from the fixed space of the mentioned collection to the space L∞. Solution to these problems became possible due to the introduction of the notion a the canonical graph of an arbitrary distribution.