Research Interests

 

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I am a student of approximation theory. My research interests are

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univariate splines :

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both polynomial

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and generalized, such as

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L-splines,

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Chebyshevian splines (very general ones - with different sections possibly in different Extended Chebyshev spaces),

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splines with connection matrices (very general ones - not necessarily totally positive);

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and also some wavelets.


Two main questions in spline theory that I study are

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de Boor's problem both for polynomial and for generalized splines;

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the connection between

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existence of blossoms,

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existence of B-splines,

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and Schoenberg-Whitney type interpolation theorems

for spaces of generalized splines (such as Chebyshevian splines with connection matrices).

Find more detail on these in two sections below. Also see a section with spline-related links.

De Boor's problem

De Boor's problem consists in ascertaining the existence of mesh-independent bounds for L-norms of orthogonal projectors onto various spline spaces (also referred to as Lebesgue constants of these spaces) and in estimating these bounds as accurately as possible in case they exist.

The importance of the problem is threefold :

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Studying the norms of orthogonal projection operators onto various approximating spaces (i.e. their Lebesgue constants) is a classical problem in approximation theory, important in its own right. In classical approximation theory, Lebesgue constants of trigonometric Fourier series and of Fourier series in various orthogonal polynomials have been and still are a subject of extensive research. In spline theory, the orthogonal projection operator has been studied since 1963 and there still remains much to be done. The study of Lebesgue constants for splines is thus an interesting problem in itself.

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The operator of orthogonal projection onto splines of order k is closely related to the operator of interpolation by splines of order 2k. In fact, the study of Lebesgue constants was initiated by Carl de Boor with the aim of deriving new results for interpolation operators. More precisely, the derivation of estimates for the norms of orthogonal projection operators leads to estimates for the error of approximation by interpolating splines, an approximation apparatus which is extensively used both for theoretical purposes and for the solution of practical approximation problems of all kinds.

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Finally, constructive estimates for Lebesgue constants can be useful in deriving and studying the behaviour of various practical algorithms of approximation by splines, such as:
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the least-squares method for the solution of boundary value problems,

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methods for the solution of integral equations.

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Estimates for the max-norm of the inverse of the B-spline Gramian matrix lead to upper bounds on the size of B-spline coefficients of a function's orthogonal projection onto the space of splines. This knowledge is useful if one were to numerically compute the orthogonal projection.

History.

My results.

Equivalent conditions for generalized spline spaces

This is joint work with Marie-Laurence Mazure, my French scientific supervisor. It was done in 2005 during my four-month visit to the Laboratoire de Modélisation et Calcul (LMC) of Université Joseph Fourier (Grenoble, France), which was sponsored by an INTAS Young Scientist Fellowship Grant (ref. no. 04-83-2601).

Its importance.

History.

My results.

Spline resources on the web

Here are some useful resources related to splines.

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home pages of spline researchers :
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Carl de Boor - one of the fathers of spline theory, author of a definitive book on splines ("A Practical Guide to Splines" - Springer, 1978; revised edition, 2001). His site contains links to a large number of his articles, both newer and older ones, many of which were fundamental for the development of spline theory. A major theme in his recent research is multivariate polynomial interpolation schemes.

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Larry Schumaker - one of the fathers of spline theory, author of a definitive book on splines ("Spline Functions: Basic Theory" - Wiley, 1981; reprint edition, Krieger, 1993). His site contains links to his recent articles (since 1995). The main subject of his recent research is constructing bi- and tri-variate spline macro-elements.

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Yuri Nikolaevich Subbotin (page in Russian, page in English) - my Russian scientific supervisor, a leading researcher in spline theory and finite-element methods in Russia, has done much to make both of these known and popularize them in Russia; translator into Russian of the first book on splines ("The Theory of Splines and Their Applications" J.H. Ahlberg, E.N. Nilson, J.L. Walsh - Academic Press, 1967; Russian edition, Mir Publishers, 1972), co-author of the first book on splines by Russian authors ("Splines in Numerical Mathematics" S.B Stechkin, Yu.N. Subbotin - Nauka, 1978). The main subjects of his recent research are :
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construction of analytic and harmonic wavelets for boundary value problems,

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approximative and extremal properties of polynomial splines and L-splines,

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extremal problems of interpolation and interpolation in the mean,

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exact calculation of n-widths,

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optimal recovery of functions from incomplete information,

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optimal bases and error bounds for the finite-element method.

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Marie-Laurence Mazure - my French scientific supervisor. The main subject of her recent research is blossoms and their applications to splines. She has been very prolific in this area, with over 30 publications on this topic in the recent years! Her site, however, does not contain links to her publications, only their names, and is out-of-date.

all the rest in alphabetical order :
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Alexei Shadrin. This is his old site at the Institut für Geometrie und Praktische Mathematik in Aachen, where he no longer works. It contains links to his most important articles on spline theory, including those on the B-spline basis condition numbers and on de Boor's problem. This is the site of the Cambridge Numerical Analysis Group, where he works now, but he has no personal web-page at this site.

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Charles Chui - His site contains links to his recent articles (since 2000).

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Frank Zeilfelder - His site contains links to his articles, except for the most recent ones.

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Gerald Farin - His site contains links to his recent articles, syllabi of his computer graphics courses, and a real wealth of useful computer graphics-related links.

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Günter Mühlbach - His site, however, contains little information and has only the names of his recent articles (since 2000).

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Günter Nürnberger - His site contains the names of all his articles, but has links only to a few of them.

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Karl Scherer - His site has no information on his articles, but instead has links to his useful lecture notes on splines and wavelets.

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Knut Mørken - His site is under construction, but already has the names of all his articles and links to the recent ones (for now, this is since 2005), as well as to the courses he is teaching.

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Marian Neamtu - His site contains the names of all his publications and has links to his recent articles (since 1993).

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Michael Floater - His site contains links to his recent articles and technical reports (since 1997).

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Michael Unser - His site contains a wealth of information both on him and on his research group: links to all of their articles, tutorials, reviews, demos, algorithms, and so on.

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Ming-Jun Lai - His site contains a wealth of information: links to his articles, information on his work on splines, wavelets, PDEs, CAGD, and GeoMath.

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Peter Alfeld - His site contains a wealth of information: links to his recent articles and talks (and also links to some older ones), interactive applets that allow one to explore the Bernstein-Bézier form of bi- and tri-variate polynomials, and other things.

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Peter Oswald - His site contains links to his recent articles (since 1996).

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Ronald DeVore - His site contains the names of all his articles, but has links to only a few of them.

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Tim Goodman - His site contains links to most of his recent articles.

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Tom Lyche - His site contains links to his recent articles (since 1997), as well as to the courses he is teaching.

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spline bibliography - a comprehensive bibliography of articles on splines and on related subjects (begun by L.L. Schumaker and currently kept up by C. de Boor); searchable and very useful

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spline applets :
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"Interpolation with polynomials and splines" - an instructive applet that allows the comparison of polynomial and natural cubic spline interpolants to the same data points. One can add, delete, and move data points and thus compare the two interpolants and examine their behaviour as the points are being moved. In mathematics, too, a picture is often worth a thousand words!

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"Connecting the dots" - not exactly an applet, but rather a collection of Matlab and Mathematica files from a colloquim on polynomial and spline interpolation. Interesting examples are discussed and plotted.

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Hartmut Prautzsch applets. This is an "interactive tutorial on geometric modeling". It is a collection of many applets illustrating different topics in GAGD.

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Gerald Farin applets. Those are the applets corresponding to the illustrations in Gerald Farin's book "Curves and Surfaces for CAGD - A Practical Guide" (5th edition, Academic Press, 2002).

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Peter Alfeld applets. This is an applet that allows one to explore the Bernstein-Bézier form of bivariate polynomials. This does the same in the trivariate case.

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This site was last updated 07-12-2006.