# 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. Kyoto Journal of Mathematics (2019) 59 (4): 869-895.
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.