De Boor's problem
De Boor's problem consists in ascertaining
the existence of meshindependent 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 :

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. 

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. 

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:

the leastsquares method for the solution of boundary value problems, 

methods for the solution of integral equations. 

Estimates for the maxnorm
of the inverse of the Bspline Gramian matrix lead to upper bounds on the
size of Bspline 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. 
Spline resources on the web
Here are some
useful resources related to splines.

home pages of spline
researchers :

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. 

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 trivariate spline
macroelements. 

Yuri Nikolaevich Subbotin (page in
Russian, page in
English)  my
Russian scientific supervisor, a leading researcher in spline theory and
finiteelement 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), coauthor 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 :

construction of analytic and harmonic wavelets for boundary value
problems, 

approximative and extremal properties of polynomial
splines and Lsplines, 

extremal problems of interpolation and interpolation in
the mean, 

exact calculation of nwidths, 

optimal recovery of functions from incomplete
information, 

optimal bases and error bounds for the finiteelement
method. 


MarieLaurence 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
outofdate. 
all
the rest in alphabetical order :

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 Bspline
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 webpage at this site. 

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

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

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

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

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

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

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. 

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

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

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. 

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

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 BernsteinBézier form of
bi and trivariate polynomials, and other things. 

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

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

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

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


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 

spline applets :

"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! 

"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. 

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

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). 

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

