Nonstandard analysis and generalized function theory

01 January 2018 → 31 December 2018
Research Foundation - Flanders (FWO)
Research disciplines
No data available
Nonstandard analysis
Project description

This project aims to further develop the nonstandard theory of generalized functions. Linear

generalized functions, commonly called (Schwartz) distributions, based on the method of duality,

are naturally represented in nonstandard analysis as smooth hyperreal functions, on which the

same operations can be defined as on usual smooth functions. Smooth hyperreal functions thus

provide a model for physical phenomena in which the model involves operations that are ill-defined

for distributions. In this sense, they share many properties with nonlinear generalized functions in

analysis. We will develop new intrinsic notions for hyperreal functions which reduce to familiar

notions for distributions. In particular, we want to develop a regularity theory for smooth hyperreal

functions different from the regularity theory in existing nonlinear theories of generalized functions,

and a corresponding theory of microlocal regularity, with the goal to qualitatively describe the

location of singularities of solutions to partial differential equations as they propagate in time, in

cases where distribution theory falls short.