# Research

My research is about generalizations of the concept of functions (say,
from R^n to C) in the frameworks of functional analysis, nonstandard analysis
and algebras of generalized functions. More specifically, the generalized
functions in these frameworks generalize (and often contain a copy of)
the space of Schwartz distributions, familiar to physicists.
### Nonstandard analysis (NSA)

Nonstandard analysis is a rigorous foundation for the infinitesimal calculus
used by mathematicians and physicists since the early days of mathematical
analysis, but later rejected and replaced by Weierstrass' "epsilon
delta"-approach, exactly because of the lack of rigour.
NSA was developed in the 1960's by the logician Abraham Robinson as a
direct application of model theory. The use of infinitesimal quantities brings
with it significant simpliciations, intuitions and facilitation for invention
in analysis.
### Schwartz distributions

Schwartz distributions are an example of a generalization of the concept of
functions. They originated from physics and contain the Dirac-delta
distribution, which is zero outside the origin and for which
the "area under the curve" is nevertheless equal to one (no classical
function can satisfy these requirements). They are a particularly powerful tool
in the study of linear partial differential equations. Some natural operations
on functions (like the product) cannot be naturally defined on arbitrary
distributions, which makes them sometimes less suited for nonlinear equations.
Although different mathematical foundations for the distributions were developed
in the 1950's, the functional analytic approach of Laurent Schwartz (which
earned him a Fields Medal in 1950) is most widely used.
### Algebras of generalized functions

Algebras of generalized functions (introduced under the impetus of the
French mathematician J.-F. Colombeau) are a more recent approach to
generalized functions than distributions. Distributions can in some sense
be viewed as special cases. Moreover, a lot of operations that are not
well-defined on distributions (like multiplication) are well-defined on
the algebras. They find their main application in the study of partial
differential equations with singular data or coefficients and in geometric
problems, e.g. in General Relativity. The algebras are similar to algebras
of nonstandard functions, and their structure is clarified by nonstandard
principles in a "cheap" version of nonstandard analysis (as
Terence Tao calls it).
### Publications

C. Impens, H. Vernaeve, Asymptotics
of differentiated Bernstein polynomials, Constr. Approx. (2001) 17: 47-57.
H. Vernaeve,
Optimal embeddings of distributions into algebras,
Proc. Edinburgh Math. Soc. (2003) 46: 373-378.
H. Vernaeve, Embedding
distributions in algebras of generalized functions
with singularities, Monatshefte fuer Mathematik (2003) 138: 307-318.
H. Vernaeve, Group
invariant Colombeau generalized functions,
Monatshefte fuer Mathematik (2008) 153: 165-175.
C. Hanel, E. Mayerhofer, S. Pilipovic, H. Vernaeve, Homogeneity in
generalized function algebras, Journal of Mathematical Analysis
and Applications (2008) 339: 889-904.
M. Oberguggenberger, H. Vernaeve, Internal sets and
internal functions in
Colombeau theory, Journal of Mathematical Analysis and Applications
(2008) 341: 649-659.
H. Vernaeve,
The local structure of nonstandard representatives
of distributions, Portugaliae Mathematica (2008) 65: 321-337.
T. Todorov, H. Vernaeve, Full
algebra of generalized functions and non-standard
asymptotic analysis, Logic and Analysis (2008) 1: 205-234. Also available as *open access* on the website of the Journal of Logic and Analysis.
H. Vernaeve, Pointwise
characterizations in generalized function algebras,
Monatshefte fuer Mathematik (2009) 158: 195-213.
H. Vernaeve, Ideals in the ring
of Colombeau generalized numbers,
Communications in Algebra (2010) 38 (6): 2199-2228.
H. Vernaeve, Weak homogeneity in
generalized function algebras, Mathematische Nachrichten (2010) 283 (10):
1506-1522.
C. Garetto, H. Vernaeve, Hilbert
˜C-modules: structural
properties and applications to variational problems, Transactions of
the American Mathematical Society (2011) 363: 2047-2090.
H. Vernaeve, Isomorphisms of
algebras of generalized functions, Monatshefte
fuer Mathematik (2011) 162: 225-237.
H. Vernaeve, Nonstandard
principles for generalized functions, Journal of Mathematical
Analysis and Applications (2011) 384: 536-548.
H. Vernaeve, J. Vindas, Characterization of distributions having a value
at a point in the sense of Robinson, Journal of Mathematical Analysis and Applications (2012) 396(1): 371-374.
E. Allaud, A. Delcroix, V. Dévoué, J.-A. Marti, H. Vernaeve, Paradigmatic wellposedness
in some generalized characteristic Cauchy problems, In: Proceedings of the 8th congress of the International Society for Analysis, its Applications, and Computation (ISAAC), Moscow, Russia, August 22-27, 2011. (2013) 476-489. Moscow: People's Friendship University of Russia.
A. Khelif, D. Scarpalézos, H. Vernaeve, Asymptotic ideals (ideals in the ring of Colombeau generalized
constants with continuous parametrization), Communications in Algebra (2014) 42 (6): 2721-2739.
H. Vernaeve, Topological properties of regular generalized function algebras, Monatshefte fuer Mathematik (2014) 173: 433-439.
H. Vernaeve, J. Vindas, A. Weiermann, Asymptotic distribution of integers with certain prime factorizations, Journal of Number Theory (2014) 136: 87-99.
P. Giordano, M. Kunzinger, H. Vernaeve, Strongly internal sets and generalized smooth functions, Journal of Mathematical Analysis and Applications (2015) 422: 56-71.
H. Vernaeve, Microlocal analysis in generalized function algebras based on generalized points and generalized directions,
Monatshefte fuer Mathematik (2016) 181: 205-215.
H. Vernaeve, An application of internal objects to microlocal analysis in generalized function algebras. In: M. Oberguggenberger, J. Toft, J. Vindas, P. Wahlberg (eds), Generalized Functions and Fourier Analysis, Operator Theory: Advances and Applications vol.260, 2017, Birkhauser, 237-251.
V. Dévoué, J.-A. Marti, H. Vernaeve, J. Vindas, Generalized functions on the closure of an open set. Application to uniqueness of some characteristic Cauchy problem, Novi Sad J. Math. (2016) 46 (2): 163-180.
A. Debrouwere, H. Vernaeve, J. Vindas, Optimal embeddings of ultradistributions into differential algebras. Monatshefte fuer Mathematik (2018) 186: 407-438.
A. Debrouwere, H. Vernaeve, J. Vindas, A non-linear theory of infrahyperfunctions. *To appear* in Kyoto Journal of Mathematics.
H. Vernaeve, Regularity of nonlinear generalized functions: a counterexample in the nonstandard setting. Preprint on *arXiv*.
*Disclaimer.* Preprints on arXiv are the latest drafts of the
papers. Substantial differences with the published
version may exist.