Research

My research is about generalizations of the concept of functions (say, from R^n to C) in the frameworks of nonstandard analysis and of Colombeau generalized functions. More specifically, these frameworks are suitable to overcome some of the limitations of the distribution-approach, 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.

Distributions

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). Some operations on functions can be defined also on distributions (like addition, product with a smooth function, differentiation, convolution, ...), but others cannot without losing essential properties (like the product of two distributions, or more general nonlinear operations on distributions). Although different mathematical foundations for the distributions were developed in the 1950's, the functional analytic approach of Laurent Schwartz is most widely used.

Colombeau generalized functions

Colombeau generalized functions (named after 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 Colombeau generalized functions. They find their main application in the study of partial differential equations. Colombeau theory is an active field of research.

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.
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, Submitted.
E. Allaud, A. Delcroix, V. Dévoué, J.-A. Marti, H. Vernaeve, Paradigmatic wellposedness in some generalized characteristic Cauchy problems, Proceedings of the ISAAC Conference 2011, To appear.

Disclaimer. Preprints on arXiv are the latest drafts of the papers. Substantial differences with the published version may exist.