General Tauberian remainder theory

Tauberian theory deals with the problem of obtaining asymptotic information of an object
(typically a function) from an 'average' (transform) of the function. The goal of the proposal is to
further develop Tauberian theory in general and Tauberian remainder theory in particular. The
project mainly focuses on three chief topics in Tauberian remainder theory.
1) The Wiener-Ikehara and Ingham-Karamata theorems: these theorems are two cornerstones in
Tauberian theory because of their many applications in different fields. Finding remainder versions
of them is currently a mainstream research topic.
2) General Tauberian remainder theorems: these kind of theorems provide Tauberian results for a
very large class of transforms at once. Because of the generality of the transforms, the potential
for applications is huge. However, up until now, no 'useful' general Tauberian remainder results
are known. This part of the project envisages breakthrough research, as it intends to provide a
general Tauberian theory, suited for applications in several branches of mathematics.
3) Remainder Tauberian theorems for large Laplace transforms: in partition theory, a branch of
combinatorics, one often encounter functions with large Laplace transforms. Up to now, Tauberian
theory is only able to deduce the main term of the asymptotic expansion from the large Laplace
transform. The project intends to provide Tauberian theorems which give more precise
information.