Nonstandard principles for generalized functions

TL;DR

This paper introduces nonstandard principles to simplify and clarify the nonlinear theory of generalized functions, aiming to reduce reliance on direct net manipulations and enhance understanding.

Contribution

It presents a tutorial on applying nonstandard principles in generalized function theory, highlighting their role in solving problems more efficiently.

Findings

Nonstandard principles can streamline proofs in generalized functions.

These principles help clarify complex nonlinear problems.

Tutorial demonstrates practical application of nonstandard methods.

Abstract

Someone knowledgeable in nonstandard analysis may get the feeling that in the nonlinear theory of generalized functions, too often one works directly on the nets and spends effort to obtain results that should be clear from general principles. We want to show that such principles can indeed be introduced and to illustrate their role to solve problems. This text is intended as a tutorial on the use of nonstandard principles in generalized function theory intended for researchers in the nonlinear theory of generalized functions.