The files are now also available in

If you have trouble because of file contamination, specify

The files are in order of increasing age.

On the (bi)infinite case of Shadrin's theorem concerning the $L_\infty$-boundedness of the $L_2$-spline projector;

Carl de Boor

july 2011

has appeared in the Subbotin 75th anniversary volume

Trudy Instituta Matematiki i Mekhaniki UrO RAN; 17(3); 2011, 25--29;

On Radon's recipe for choosing correct sites for multivariate polynomial interpolation;

Dominik Stahl and Carl de Boor

june 2011

\JAT; 163(12); 2011; 1854--1858;

The way things were in multivariate splines: A personal view;

Carl de Boor

april 2009

in Multiscale, Nonlinear and Adaptive Approximation, R. DeVore and A. Kunoth

(eds.), Springer Verlag (Berlin-Heidelberg, Germany); 2009; 19--37;

Multivariate polynomial interpolation: Aitken-Neville sets and generalized principal lattices;

Carl de Boor

april 2008

updated 1may08, 8jul08, 10aug08

\JAT; 161(1); 2009; 411--420;

On the pointwise limits of bivariate Lagrange projectors;

Carl de Boor and Boris Shekhtman

november 2007

\LAA; 429(1); 2008; 311--325;

Box splines revisited: convergence and acceleration methods for the subdivision and the cascade algorithms;

Carl de Boor and Amos Ron

december 2006

\JAT; 150(1); 2008; 1--23;

Multivariate polynomial interpolation: conjectures concerning GC-sets;

Carl de Boor

september 2006

proc. 1st Dolomites Workshop: 2006:

07dec06 update

\NA; 45; 2007; 113--125;

What are the limits of Lagrange projectors?

Carl de Boor

december 2005

(Constructive Theory of Functions, Varna 2005), B. Bojanov (ed.),

Marin Drinov Acad.\ Publ.\ House (Sofia); 2006; 51--63;

Ideal interpolation: Mourrain's condition vs $D$-invariance;

Carl de Boor

jan 2005

(Banach Center Publications Vol.~72: Approximation and Probability),

Tadeusz Figiel and Anna Kamont (eds.), IMPAN (Warszawa, Poland); 2006; 49--55;

Ideal interpolation;

Carl de Boor

dec 2004

\TexasXI; 59--91;

dec05: three footnotes added

Divided differences;

Carl de Boor

nov 2004

Surveys in Approximation Theory; 1; 2005; 46--69;

http://www.math.technion.ac.il/sat/papers/2

Interpolation from spaces spanned by monomials

Carl de Boor

aug 2004

\AiCM; 26(1-3); 2007; 63--70;

An efficient definition of the divided difference

Carl de Boor

aug 2004

in (Approximation Theory: A Volume Dedicated to Borislav Bojanov),

D. K. Dimitrov, G. Nikolov, and R. Uluchev (eds.), Marin Drinov Academic

Publ. House (Sofia); 2004; 58--63;

On interpolation by radial polynomials

Carl de Boor

dec 2003

as of 21jul04/23feb04

\AiCM; 24; 2006; 143--153;

An asymptotic expansion for the error in a linear map that reproduces polynomials of a certain order

Carl de Boor

apr 2003

\JAT; 134; 2005; 171--174;

The B-spline recurrence relations of Chakalov and of Popoviciu

Carl de Boor and Allan Pinkus

mar 2003

\JAT; 124(1); 2003; 115--123;

A divided difference expansion of a divided difference

Carl de Boor

nov 2002

\JAT; 122(1); 2003; 10--12;

A Leibniz formula for multivariate divided differences

Carl de Boor

apr 2002

\SJNA; 41(3); 2003; 856--868;

Spline Basics

Carl de Boor

dec 2000

Chapter 6 in \Cagdhand; see (cagd.snu.ac.kr/main.html)

What is the inverse of a basis?

Carl de Boor

sep 2000

BIT; 41(5); 2001; 880--890;

(the printed version contains a random 7 right in the middle of (4.2), the most

important formula of the paper, and an extra p in the first display on page 889)

Polynomial interpolation to data on flats in $\Rd$

Carl de Boor, Nira Dyn, Amos Ron

aug 1999

\JAT; 105; 2000; 313--343;

On Ptak's derivation of the Jordan normal form;

Carl de Boor

may 1999

\LAA; 310; 2000; 9--10;

Computational aspects of multivariate polynomial interpolation: Indexing the coefficients

Carl de Boor

jan/feb 1999

may99 incorporated referee's comments

\AiCM; 12; 2000; 289--301;

Calculation of the smoothing spline with weighted roughness measure

Carl de Boor

sep 1998

Math.\ Models Methods Appl.\ Sci.; 11(1); 2001; 33--41;

Multivariate Hermite interpolation (talk at Guernavaca, 13apr99)

Carl de Boor

apr 1999

updated 08feb01 to conform to present-day notation

updated 20apr08 to correct some misprints

A selfunwrapping wrapper containing MATLAB 5 programs for the construction and evaluation of the least interpolant to data in any number of dimensions

Carl de Boor

as of February 1999

The error in polynomial tensor-product, and in Chung-Yao, interpolation

Carl de Boor

February 1997

in

Surface Fitting and Multiresolution Methods

A. Le M\'ehaut\'e, C. Rabut, and L. L. Schumaker (eds),

Vanderbilt University Press (Nashville TN), 35--50.

6apr09: added missing reference to Waldron'98a

On the Meir/Sharma/Hall/Meyer analysis of the spline interpolation error

Carl de Boor

December 1996

Appeared in:

\Powellfest; 47--58;

Approximation orders of FSI spaces in $L_2(\Rd)$

Carl de Boor, Ron DeVore, and Amos Ron

March 1996

additional references added June-July 1996

referees' comments incorporated Aug/Sep 1996

a minor but useful variation of main result and add'l refs added feb97

An earlier version was, unfortunately, printed as

\CA; 14; 1998; 411--429;

The correct version is

\CA; 14; 1998; 631--652;

The multiplicity of a spline zero

Carl de Boor

December 1995

January 96 (reflect referee's comments)

appeared in

\AoNM; 4; 1997; 229--238;

On determining the foot of the continental slope

Carl de Boor

November 1995

revised feb'96

On ascertaining inductively the dimension of the joint kernel of certain commuting linear operators. II

Carl de Boor, Amos Ron, Zuowei Shen

May 1995

Adv. in Math. {\bf 123} (1996), 223--242.

On the Sauer-Xu formula for the error in multivariate polynomial interpolation;

Carl de Boor

March 1995

\MC; 65; 1996; 1231--1234;

A multivariate divided difference

Carl de Boor

March 1995

\TexasVIIIa; 87--96;

% 25mar02: supplied missing g in (4.1) and completed refs [17] and [18].

Gauss elimination by segments and multivariate polynomial interpolation

Carl de Boor

March 1994

in (Approximation and Computation), R.V.M. Zahar (ed.),

ISNM 119, Birkh\"auser Verlag (Basel-Boston-Berlin); 1994; 1--22;

On the evaluation of box splines,

Carl de Boor

March 1993

has appeared in Numer.Algorithms; 5; 1993; 5--23;

an alternative approach, particularly good for multiple directions, can be

found in Leif Kobbelt's paper

Stable evaluation of box splines;

\NA; 14(4); 1997; 377--382;

% available at

% http://www.mpi-sb.mpg.de/~kobbelt/papers/boxeval.ps.gz

% with the corresponding set of m-files at

% http://www.mpi-sb.mpg.de/~kobbelt/papers/boxeval.tgz

Multivariate piecewise polynomials,

Carl de Boor

October 1992

has appeared in Acta Numerica; 2; 1993; 65--109;

Approximation order without quasi-interpolants

Carl de Boor

August 1992

has appeared in \TexasVII; 1--18;

On ascertaining inductively the dimension of the joint kernel of certain commuting linear operators

Carl de Boor, Amos Ron, Zuowei Shen

June 1992

updated Apr 96 to reflect copy editor's changes

has appeared in Adv. Applied Math; 17; 1996; 209--250;

A sharp upper bound on the approximation order of smooth bivariate pp functions

Carl de Boor and Rong-Qing Jia

March 1992

has appeared in J.Approx.Theory; 72(1); 1993; 24--33;

On the construction of multivariate (pre)wavelets

Carl de Boor, Ronald A. DeVore, Amos Ron

February 1992

has appeared in Constr.Approx.; 9; 1993; 123--166;

The structure of finitely generated shift-invariant spaces in $L_2(\RR^d)$

Carl de Boor, Ronald A. DeVore, Amos Ron

February 1992

has appeared in J. of Functional Analysis 119(1); 1994; 37--78;

% minor correction in intro 01feb01

On the error in multivariate polynomial interpolation

Carl de Boor

1991

has appeared in Applied Numerical Mathematics; 10; 1992; 297--305;

Approximation from shift-invariant subspaces of $L_2(\RR^d)$

Carl de Boor, Ronald A. DeVore, Amos Ron

July 1991

has appeared in Trans.Amer.Math.Soc. 341; 1994; 787--806;

% note, this file has the name `ell-2-shift', not `one-two-shift'.

% misprint corrected: 19oct94, 17nov97

Fourier analysis of the approximation power of principal shift-invariant spaces

Carl de Boor, Amos Ron

July 1991

has appeared in Constr.Approx.; 8; 1992; 427--462;

Quasiinterpolants and approximation power of multivariate splines

Carl de Boor

July 1990

has appeared in

(Computations of curves and surfaces), Dahmen, Gasca, Micchelli

(eds.), Kluwer (Dordrecht, Netherlands); 1990; 313--345;

The least solution for the polynomial interpolation problem;

Carl de Boor, Amos Ron

June 1990

has appeared in Math.Zeitschrift; 210; 1992; 347--378;

Computational aspects of polynomial interpolation in several variables

Carl de Boor, Amos Ron

March 1990

has appeared in Math.Comp.; 58; 1992; 705--727;

An empty exercise

Carl de Boor

March 1990

has appeared in

ACM SIGNUM Newsletter; 25(4); 1990; 2--6;

Polynomial interpolation in several variables

Carl de Boor

March 1990

has appeared in

(Studies in Computer Science {(in Honor of Samuel D. Conte)}),

R. DeMillo and J. R. Rice (eds.), Plenum Press (New York); 1994; 87--119;

An alternative approach to (the teaching of) rank and dimension

Carl de Boor

February 1990

has appeared in

\LAA; 146; 1991; 221--229;

The exponentials in the span of the multiinteger translates of a compactly supported function: quasiinterpolation and approximation order

Carl de Boor and Amos Ron

November 1989

has appeared in

J. London Math. Soc. (2); 45; 1992; 519--535;

Polynomial ideals and multivariate splines

Carl de Boor, Amos Ron

June 1989

has appeared in

(Multivariate Approximation Theory IV, ISNM 90),

C. Chui, W. Schempp, and K. Zeller (eds.),

Birk\-h\"auser Verlag (Basel); 1989; 31--40;

On two polynomial spaces associated with a box spline

Carl de Boor, Nira Dyn \& Amos Ron

April 1989

has appeared in

\PJM; 147; 1991; 249--267;

% (may'99: corrected and updated the references)

On polynomial ideals of finite codimension with applications to box spline theory;

Carl de Boor, Amos Ron

December 1988

has appeared in

\JMAA; 158; 1991; 168--193;

% (de.01: corrected (3.8), changed pi to Pi, updated one reference)

Fundamental solutions for multivariate difference equations;

Carl de Boor, Klaus H\"ollig, Sherman Riemenschneider

1988

has appeared in

\AJM; 111; 1989; 403--415;

On multivariate polynomial interpolation

Carl de Boor, Amos Ron

November 1988

has appeared in Constr. Approx.; 6; 1990; 287--302;

% (13nov98: corrected two misprints and changed the symbol for polynomial

% space from \pi to \Pi)

The limit at the origin of a smooth function space;

Carl de Boor, Amos Ron

November 1988

has appeared in \TexasVI; 93--96;

Local corner cutting and the smoothness of the limiting curve;

Carl de Boor

November 1988

has appeared in \CAGD; 7; 1990; 389--397;

The exact condition of the B-spline basis may be hard to determine

Carl de Boor

July 1988

has appeared in \JAT; 60; 1990; 344--359;

What is a multivariate spline?

Carl de Boor

August 1987

has appeared

(Proc.\ First Intern.\ Conf.\ Industr.\ Applied Math., Paris 1987), J.\

McKenna and R. Temam (eds.), SIAM (Philadelphia PA); 1988; 90--101;

The condition of the B-spline basis for polynomials

Carl de Boor

April 1987

has appeared as

\SJNA; 25(1); 1988; 148--152;

but the journal's final computer-aided processing messed up the paper through

the omission of various words (that happened to lie beyond column 80 on a

line in the TeX file).

B-spline basics

Carl de Boor

MRC 2952, 1986

in (Fundamental Developments of Computer-Aided Geometric Modeling),

Les Piegl (ed.), Academic Press (London) 1993; 27--49;

% Corrected (in Section 12) on 04 mar 96.

% Scaling of figures adjusted and misprints corrected on 03 jun 96

% A misprint corrected (and adjusted to current tex-macros) on 06 jun 96

% A misprint corrected on 12feb98

% A reference updated 27apr09 but the change to (2.4b) made then rescinded on 30mar10

Multivariate Approximation

Carl de Boor

MRC 2950, August 1986

in (State of the Art in Numerical Analysis),

A. Iserles and M. Powell (eds.),

Institute Mathematics Applications (Essex); 1987; 87--109;

% 10mar09: some references, some notation (for pol.spaces, intervals, etc)

% updated; unfortunately, the picture files are lost.

$B$--form basics;

Carl de Boor

1986

in (Geometric Modeling: Algorithms and New Trends),

G. E. Farin (ed.),

SIAM Publications (Philadelphia); 1987; 131--148;

% 30nov09 supplied the missing Figure 3, and corrected a typo in (3.3) and in

% last display before Section 6.

The polynomials in the linear span of integer translates of a compactly supported function

Carl de Boor

1986

has appeared in: \CA; 3; 1987; 199--208;

% may99: updated the references, corrected various typos, and made the changes:

% \Pi --> \tilde\Pi, \pi -->\Pi, scale-invariant --> dilation-invariant,

% scale (S_h) --> ladder (S_h)

Convergence of cardinal series;

Carl de Boor, Klaus H\"ollig, Sherman Riemenschneider

August 1985

has appeared: \PAMS; 98(3); 1986; 457--460;

Convergence of cubic spline interpolation with the not-a-knot condition

Carl de Boor

1985

MRC TSR 2876, October 1985

Partitions of unity and approximation

Carl de Boor and Ronald DeVore

May 1984

has appeared in \PAMS; 93(4); 1985; 705--709;

A geometric proof of total positivity for spline interpolation

Carl de Boor and Ron DeVore

March 1984

has appeared in \MC; 45(172); 1985; 497--504;

Convergence of bivariate cardinal interpolation;

C. de Boor, K. H\"ollig, S. Riemenschneider

1984

has appeared in \CA; 1; 1985; 183--193;

Controlled approximation and a characterization of the local approximation order

Carl de Boor and R.-Q. Jia

1984

has appeared in Proc.\ AMS; 95(4); 1985; 547--553;

http://www.ams.org/journals/proc/1985-095-04/S0002-9939-1985-0810161-X/S0002-9939-1985-0810161-X.pdf

Bivariate cardinal interpolation by splines on a three-direction mesh;

Carl de Boor, Klaus H\"ollig, Sherman Riemenschneider;

1984

\IJM, 29(4), 1985, 533--566;

Approximation order from bivariate $C^1$-cubics: A counterexample

Carl de Boor and Klaus H"ollig

1982

has appeared as

\PAMS; 87(4); 1983; 649--655;

Approximation order from smooth bivariate splines

C. de Boor, R. DeVore, K. H\"ollig

1982

has appeared in TexasIV; 353--357;

Topics in multivariate approximation theory

C. de Boor

May 1982

TSR #2379, Mathematics Research Center, University of Wisconsin-Madison

Approximation by smooth multivariate splines

Carl de Boor and Ron DeVore

December 1981

has appeared in \TAMS; 276(2); 1983; 775--788;

Inverses of infinite sign regular matrices

C. de Boor, S. Friedland, and A. Pinkus

November 1980

has appeared: \TAMS; 274(1); 1982; 59--68;

Recurrence relations for multivariate B-splines

Carl de Boor and Klaus H\"ollig

May 1981

has appeared: \PAMS; 85(3); 1982; 397--400;

The inverse of a totally positive bi-infinite band matrix

C. de Boor

November 1980

has appeared: \TAMS; 274(1); 1982; 45--58;

Local piecewise polynomial projection methods for an O.D.E.\ which give high-order convergence at knots

Carl de Boor and Blair Swartz

summer 1980

has appeared in \MC; 36; 1981; 21--33;

On a max-norm bound for the least-squares spline approximant

Carl de Boor

1980

has appeared in

(Approximation and Function Spaces),

C. Ciesielski (ed.),

North Holland (Amsterdam); 1981; 163--175;

Collocation approximation to eigenvalues of an ordinary differential equation: Numerical illustrations;

Carl de Boor and Blair Swartz

March 1980

has appeared in \MC; 36(153); 1981; 1--19;

Collocation approximation to eigenvalues of an ordinary differential equation: The principle of the thing;

Carl de Boor and Blair Swartz

March 1979

has appeared in \MC; 35(151); 1980; 679--694;

Efficient Computer Manipulation of Tensor Products:

Carl de Boor

oct 1978

has appeared in

ACM Trans. Math. Software; 5(2); 1979; 173--182;

Final production introduced many typos, all marked in the present copy

see Corrigenda, ACM Trans. Math. Software; 5(4); 1979; 525;

How does Agee's smoothing method work?

Carl de Boor

1978

has appeared in

(Proceedings of the 1979 Army Numerical Analysis and Computers Conference),

xxx (ed.), ARO Rept.\ 79-3, Army Research Office (Triangle Park NC); 1979;

299--302;

The Numerically Stable Reconstruction of a Jacobi Matrix from Spectral Data;

Carl de Boor and G. H. Golub

apr 1977

\LAA; 21; 1978; 245--360;

Splines as linear combinations of B-splines. A Survey

Carl de Boor

1976

has appeared in \TexasII; 1--47;

% corrected version (with updated references) 19sep97

% left off an extraneous label (6.1) 01aug03

Odd-degree spline interpolation at a biinfinite knot sequence

Carl de Boor

\Tex-version of MRC TSR #1666, August 1976

has appeared in \BonnI; 30--53;

A bound on the $L_\infty$-norm of $L_2$-approximation by splines in terms of a global mesh ratio;

Carl de Boor

1975

has appeared:

\MC; 30(136); 1976; 765--771;

http://www.ams.org/journals/mcom/1976-30-136/S0025-5718-1976-0425432-1/S0025-5718-1976-0425432-1.pdf

On calculating with B-splines II. Integration;

C. de Boor, T. Lyche, L. L. Schumaker;

1975

(warning: it's a 10Mb file)

in: Numerische Methoden der Approximationstheorie Vol.\ 3, ISNM 30),

L. Collatz, G. Meinardus, and H. Werner (eds.),

Birk\-h\"auser Verlag (Basel); 1976; 123--146;

On local linear functionals which vanish at all $B$-splines but one;

Carl de Boor

1975

has appeared in (Theory of Approximation with Applications),

A. G. Law and N. B. Sahney (eds.),

Academic Press (New York); 1976; 120--145;

Cardinal interpolation and spline functions VIII: The Budan Fourier theorem for splines and applications;

Carl de Boor and I. J. Schoenberg

feb 1975

has appeared in (Lecture Notes in Mathematics 501), K. B\"ohmer (ed),

Springer-Verlag (Berlin); 1976; 1--77;

% corrected jul2000 a la KohlerNikolov95a

A smooth and local interpolant with ``small'' $k$-th derivative;

Carl de Boor

1975

has appeared in

(Numerical Solutions of Boundary Value Problems for Ordinary Differential Equations),

A. Aziz (ed.), Academic Press (New York); 1975;

177--197;

How small can one make the derivatives of an interpolating function?;

Carl de Boor

1973

has appeared as \JAT; 13; 1975; 105--116;

% corrected version (with updated references) 22oct99

On cubic spline functions that vanish at all knots;

Carl de Boor

1973

\AiM, 20(1), 1976, 1--17;

A remark concerning perfect splines;

Carl de Boor;

1973

has appeared as \BAMS; 80(4); 1974; 724--727;

http://www.ams.org/journals/bull/1974-80-04/S0002-9904-1974-13572-X/S0002-9904-1974-13572-X.pdf

Total positivity of the spline collocation matrix;

Carl de Boor;

1974

\JMM, 25(6), 1976, 724--727;

On bounding spline interpolation;

Carl de Boor

1973

\JAT; 14(3); 1975; 191--203;

% added a footnote pointing out that Jia disproved the conjecture, in %Jia88d

The quasi-interpolant as a tool in elementary polynomial spline theory;

Carl de Boor

1973

has appeared in \TexasI; 269--276;

Good approximation by splines with variable knots;

Carl de Boor

1973

has appeared in \EdmontonI; 57--72;

Subroutine Package for Calculating with B-splines;

Carl de Boor

August 1971

report LA-4728-MS Los Alamos scientific laboratory,

An extended version has appeared as

``Package for calculating with B-splines'', \SJNA, 14, 1977, 441--472.

On the approximation by $\gamma$-polynomials;

Carl de Boor

1969

has appeared in \MadisonII; 157--183;

On uniform approximation by splines

Carl de Boor

1967

has appeared as \JAT; 1; 1968; 219--235;

4oct08 various misprints have been corrected

Least Squares Cubic Spline Approximation I -- Fixed Knots

Carl de Boor, John R. Rice

CSD TR 20 April 1968

Least Squares Cubic Spline Approximation II -- Variable Knots

Carl de Boor, John R. Rice

CSD TR 21 April 1968

% error in NUBAS corrected 3oct01

On local spline approximation by moments;

Carl de Boor

Sep 1966

has appeared in

\JMM; 17; 1968; 729--735;

The method of projections as applied to the numerical solution of two point boundary value problems using cubic splines

Carl de Boor Ph.D. thesis, Univ.\ Michigan

August 1966

% corrected version (with updated references) 4apr96

On splines and their minimum properties;

Carl de Boor and Robert E. Lynch

1965

has appeared in

\JMM; 15; 1966; 953--969;

Error bounds for spline interpolation;

Garrett Birkhoff & Carl de Boor;

1964

\JMM, 13(5), 1964, 827--835;

Piecewise polynomial interpolation and approximation;

Garrett Birkhoff and Carl R. de Boor;

1965

VERY LONG FILE, 8.1Mb

has appeared in

\Generalmotors; 164--190;

Best approximation properties of spline functions of odd degree;

Carl de Boor;

1963

\JMM, 12(5), 1963, 747--750;

Bicubic spline interpolation;

Carl de Boor

1960

VERY LONG FILE, 4.8Mb

has appeared in

\JMP (J.Mathematics and Physics); 41(3); 1962; 212--218;

Viva vi!, (a brief introduction to vi)

Carl de Boor

1987

version: 03aug03

TeXnicalities (an informal introduction to TeX use) (the odd pages)

Carl de Boor

1987

version: 06feb99

TeXnicalities (an informal introduction to TeX use) (the even pages)

Carl de Boor

1987

version: 06feb99

A file of plain TeX macros useful for writing papers and books in plain TeX (including automatic sequencing of formal items and items in the bibliography, and the exact placement of items).

Carl de Boor

1983

version: 31may04

A file of plain TeX macros useful for handling the typesetting of programs and program-related material in plain TeX.

Carl de Boor

1991

version: 25feb98

A file of plain TeX macros useful for generating (the draft of) an index.

Carl de Boor

1993

version: 27jun98

TeX macros of use with the spline bibliography.

Carl de Boor

1990

version: 05jun04

TeX macros of use with the spline bibliography.

Carl de Boor

1990

version: 02may04

TeX macros of use with the spline bibliography.

Carl de Boor

1990

version: 27may04

a self-unwrapping wrapper containing files for simplifying the safe mailing of files via email.

Carl de Boor

1989

version: mar 96

List of publications of I. J. Schoenberg

(Carl de Boor)

1987

updated 10mar09