This page is intended to communicate to the scientific community about my ongoing work.
There's plenty, plenty, plenty of work to be done to really get the consequences of this out there.
However, my request for a sabatical year for research was refused, and the CNRS didn't acknowledge receipt of my latest proposal.
I experienced that before. To be honest, I'm fed up with this waste of time!
I'm even considering taking a job in the private sector (know, I really mean it. Not just an empty threat from a bitter workaholic...).
May be, in a private company, they won't pay me for wasting my time writing projects to have them refused three times before I can start implementing my new ideas 3 or 4 years from now...
Anyway, if anybody is interested, here is what I'm working on. You can find the PDF of a draft available here.
The basic 1D version is here.
The outline is:
Algebraic Framework for Mathematical and Numerical Analysis in Partially Ordered Algebras Over Z or R.
Integration Theory for Functions to Some Partially Ordered Algebras and Functional Norms
Ordinary and Symmetric Differentiation for Mixed Real-Integer Functions
Interpretation of the main convolution kernels as points [respectively derivatives] of smooth curves.
Associated Theory of Differentiability for Functions Sampling on Grids
Broad Multivariate Generalization of the Taylor Theorem With Integral Remainder, which should have broad rippling consequences on a variety of approximations results
Notion of a generalized Analytical Function (including integer only generalized power series)
Convergent integer only method method for linear PDE. Test (proof of concept) on the simple case of y' = y (exponential function)
General explicit expression of uniform B-splines as piecewise Bésier functions
Note that, at the time I post this page, I am using the time I have to think about elements of a C++ library to implement all the mathematical methods which can be defined on analyzable spaces, with high grade modern C++ generic programming. This library should also be able to link with other libraries, either from signal processing, from the exact computation communities (e.g. LinBox), or from Digital Geometry (e.g. DgTal).
In my view, there's a lot, and a lot, and a lot more to say about this... when I have a little time.