Differentiation and Integration in Several Real Variables/Test functions

From Wikibooks, open books for an open world
Jump to navigation Jump to search

We will consider two classes of test functions, namely bump functions and Schwartz functions.

Definition (Schwartz function):

Let . Then the Schwartz functions on are the set defined in the following way:


Definition (bump function):

Let . A bump function on is a function such that , the support of , is compact.

Proposition (test functions are integrable):

Let or . Then .

{{proof|Suppose first . We know that on , attains a maximum by Weierstraß' theorem.