Árpád Száz
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Convolution multipliers and distributions
1975, Pacific Journal of Mathematics
https://doi.org/10.2140/PJM.1975.60.267
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Abstract
In this paper, in a purely algebraic way, Schwartz distributions in several variables are generalized in accordance with their homomorphism interpretation proposed by R. A. Struble. 0. Introduction. R. A. Struble in [10] has shown that Schwartz distributions can be characterized simply as mappings, from the space 3) of test functions into the space % of smooth functions, which commute with ordinary convolution. This new view of distributions has turned out to be very useful [11,12] and motivated us to give a simple generalization for distributions which is closely related to Mikusiήski operators and convolution quotients of other types [11, . The method employed here is an appropriate modification of a general algebraic method . Mappings which commute with convolution are called convolution multipliers here. (Distributions can be characterized as convolution multipliers, Mikusiήski operators themselves are convolution multipliers.) In §1, convolution multipliers from various subsets of 3) into % are discussed. We are primarily concerned with their maximal extensions. In §2, a module Wl of certain maximal convolution multipliers is constructed and investigated from an algebraic point of view. In §3, Schwartz distributions are embedded and characterized in W. For example, we prove that distributions are the only continuous elements of 2)ϊ. Finally, we show that there are elements in W which are not distributions. To illustrate the appropriateness of our generalizations, we refer to the following facts: One of the difficulties in working with Schwartz distributions is that only distributions Λ satisfying Λ * 3) = 3 are invertible in 3)'. Whereas, distibutions Λ satisfying A*3) C3) such that Λ * 3) has no proper annihilators in % are invertible in 3ft. (The heat operator in two dimensions [1] seems to be a distribution which is not invertible in 3)\ but is invertible in 9K.) There are regular Mikusiήski operators [1] which are not distributions. Whereas, normal Mikusiήski operators [11] can be embedded in Wl. their maximal extensions. Let k be a fixed positive integer, R* be the kdimensional Euclidean space and C be the field of complex numbers.
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