| Abstract: | With the help of an iterative process, we define a fairly broad class of arithmetical distributions on which continuous operations of addition, multiplication, and differentiation are defined. The multiplication of arithmetical distributions that we introduce is made consistent with the known definitions of the product of distributions. The results obtained can be used to justify the passage to the limit in the study of nonlinear problems of mathematical physics. Formalizing our approach, we describe a construction of an extension of binary relations, which is called sequential extension, from a dense subset to the whole topological space. These results are extended to operations of the first order. We show that the sequential extension of differentiation from the set of infinitely differentiable functions to the set of distributions coincides up to isomorphism with the generalized differentiation of distributions. Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 836–853, June, 1999. |
