I made a Ph.D. in Measure Theory under the supervision of Prof. Richard Delanghe. After this I got fascinated by the strange universe of infinitely small and infinitely large numbers, generously offered by Infinitesimal (aka Nonstandard) Analysis. Since then my research has been devoted to analysis inside this framework.

A recurring theme is my use of hyperreal or hypercomplex polynomials of infinite degree. These functions behave (within certain limits) like ordinary polynomials but their number of terms is uncountably infinite.

These 'infinitely long' polynomials (they are infinitely much longer than any power series) provide a tool whose flexibility is unmatched in 'ordinary' analysis.

The main results regarding continuity of nonstandard polynomials can be found in

Local microcontinuity of nonstandard polynomials, Israel J.Math. 59 (1987) 81-97..
Propagation of microcontinuity for nonstandard polynomials, J.Anal.Math. 53 (1989), 187-200.

The question of how nonstandard polynomials intersect real or complex space (i.e. what they look like to the standard observer) is treated in

Real functions as traces of infinite polynomials, Math.Ann. 284 (1989) 63-73.
Standard and nonstandard polynomial approximation, J.Math.Anal.Appl. 171 (1992), 361-376.

The last paper shows that uniform continuity is a poor substitute for nonstandard (hyperreal or hypercomplex) properties. In the hyperreal framework, e.g., Weierstrass' Approximation Theorem is upgraded from a sufficient condition on compact sets to a necessary and sufficient condition on G-delta sets.

For some time now I have been trying to reduce Schwartzian distributions to nonstandard polynomials. A first approach can be found in

Representing distributions by nonstandard polynomials, Developments in nonstandard mathematics, N. Cutland, V. Neves, F. Oliveira and J. Sousa-Pinto eds. (1995), 119-129

In this paper each distribution with compact support is identified to a sequence of polynomials. Further reduction of this representation should result in identifying each distribution with compact support to a single polynomial, viz. some well defined finite-order derivative of a Bernstein polynomial of infinite order.

This reduction leads to some new problems concerning Bernstein operators. In treating these I was happy to collaborate with Heiner Gonska (Duisburg, Germany), Ioan Gavrea (Cluj-Napoca, Romania), Daniela Kacso (Cluj-Napoca, Romania) and my former Ph.D.-student, now my colleague Hans Vernaeve. See the latter's Ph.D. thesis Nonstandard Contributions to the Theory of Generalized Functions (May 30, 2002) for a comprehensive and successful treatment of the original problem along entirely different lines.

(with Hans Vernaeve) Asymptotics of differentiated Bernstein Polynomials, Constr.Approx.17 (2001), 47-57
(with Ioan Gavrea) A Leibniz differentiation formula for positive operators , J.Math.An.Appl. 271 (2002), 175-181
Dini bounds for differentiated Bernstein polynomials, J.Approx.Th. 145 (2007), 128-132

As the Ph.D. research of my assistant Sam Sanders went along, emphasis has shifted from the ultrapower approach to Elementary Recursive Nonstandard Analysis (ERNA), an axiomatic system developed in the last decade by Chuaqui, Sommer and Suppes. ERNA is a partial implementation of Hilbert's Finitistic Program. We were most happy with the expertise in logic and model theory generously offered by Ulrich Kohlenbach (Darmstadt) and Andreas Weiermann, now our colleague at Gent University.

(with Sam Sanders) Transfer and a Supremum Principle for ERNA, J. Symb. Logic 73 (2008), 689-710.
Preliminary version ERNA at Work appeared in The Strength of Nonstandard Analysis, Imme van den Berg and Vitor Neves, eds., SpringerWienNewYork 2007, pp. 64-75.

(with Sam Sanders) Saturation and $\Sigma_2$-transfer for ERNA, to appear in J. Symb. Logic 74 (2009).

(More papers submitted.)

My interest in Real Analysis, Measure and Integration is fuelled by my teaching these subjects. For some time now, I've been going into FTC (in one and several variables, both Riemann and Lebesgue). I'm fond of simple proofs of important results, as exemplified in

Stirling's series made easy, Am.Math.Monthly 110 (2003), 730-735.
Preliminary version Stirling's series for n! made easy appeared in Real Analysis Exchange, 26th Summer Symposium Conference Reports, June 2002, pp.67-71.

I also happen to give attention to a hyperdiscrete approach to (generalized) functions. Each continuous function is identified with its skeleton, defined on the hyperfinite timeline. Right and left hand derivatives are replaced with forward resp. backward divided differences with infinitesimal increment. The hypersmooth approach with suitable polynomials of infinite degree (described above) gives a tangent to everything everywhere; the hyperdiscrete approach is limited to giving right and left hand directional derivatives to everything everywhere. (To be continued.)

For some time now, I've been working on a major project with working title Mathematics, a fragment constructed from zero. It originates in my anger facing the decline of secondary school mathematics. Originally, I intended to systematically develop secondary school mathematics, as it used to be in the good old days in our finest schools. (Actually, those days turned out to be less good than I remembered.) I first called it Deductive intermediate mathematics but changed the title after switching to a constructive approach. It was never intended for secondary school students, though, and in its present form even teachers may find it hard going. It's highly "no nonsense". There are no axioms. Real numbers are defined as decimal expansions; functions like the exponential and sine are defined by their series. Everything dispensable is dispensed with. (For instance: so far, I have no need for derivatives beyond the first.) Proofs are as short as I could provide them. But everything comes with its proof. There are also no loops or loose ends; everything (so far) is deduced by pure logic. Geometry will be analytic. If you look carefully, you'll find hat most calculus books at some point rely on geometry and vice versa. This I want to avoid. The style will be austere, without external elements begging for the attention of uninterested people. No applications will be included. (You read it correctly: no applications.)