Structure Theorems for Distributions and Tempered Distributions

2023

Document Type

Thesis - Oberlin Community Only

Bachelor of Arts

Mathematics

Chris Marx

Meredith Shea

Keywords

Distribution theory, Structure theorems

Abstract

In this paper, we prove the structure theorems for the space of compactly supported distributions (E′(R)), distributions (D′(R)), and tempered distributions (S′(R)). We introduce a notion of antiderivative defined on the distribution class D′(R), and use the Riesz representation theorem to show that each element of space E′(R) can be written as a derivative of a distribution generated by a continuous function. Then, we show that the result for E′(R) may be extended to the spaces D′(R) and S′(R) with some modifications. For D′(R), we extend our conclusion about E′(R) via a partition of unity. For S′(R), we modify the definition of the distributional antiderivative, based on which we prove that each element of the space S′(R) can be written as a derivative of a tempered distribution generated by a polynomially bounded continuous function.

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