%Here are the references Carl de Boor had %For details see http://www.cs.wisc.edu/~deboor/bib/bib.html %Bos83a % sonya \refJ Bos, L.; On Kergin interpolation in the disk; \JAT; 37; 1983; 251--261; %Jia86c \refJ Jia, Rong-Qing; Extension of Kergin interpolation operators; Ke Xue Tong Bao; 31; 1986; 805--808; %Kergin78 % carl \refD Kergin, P.; Interpolation of $C^k$ Functions; University of Toronto, Canada; 1978; %Kergin80 % sonya \refJ Kergin, P.; A natural interpolation of $C^k$ functions; \JAT; 29; 1980; 278--293; %LiangXZ86 % . \refJ Liang, X. Z.; Kergin-interpolation at the points which are zeros of the bivariate polynomial of least deviation from zero on the disk (Chinese); Dongbei Shida Xueboa, (Ziran Kexue Ban), J. Northeast Normal U. of Natural Sciences, Changchun; 2; 1986; 408--414; % Shayne %LiangXZYe86 % author \refJ Liang, Xue-Zhang, Ye, Y. M.; On Kergin interpolation at the points which are zeros of the bivariate polynomial of least deviation from zero on the disk; Northeastern Math.\ J.; 4; 1986; 409--414; %Maier94 % . \refD Maier, Ulrike; Approximation durch Kergin-Interpolation; Dr., Universit\"at Dortmund (Germany); 1994; % represent interpolant in power form %Micchelli80b % larry \refJ Micchelli, C. A.; A constructive approach to Kergin interpolation in $\RR^k$: multivariate B-splines and Lagrange interpolation; \RMJM; 10; 1980; 485--497; %MicchelliMilman80 % sonya \refJ Micchelli, C. A., Milman, P.; A formula for Kergin interpolation in $\RR^k$; \JAT; 29; 1980; 294--296; %Here are the ones Shayne Waldron added %AnderssonPassare91a % shayne \refJ Andersson, M., Passare, M.; Complex Kergin interpolation; \JAT; 64; 1991; 214--225; % Another contribution to Kergin questions from complex analysts % it is shown that a C-convex domain is the most general on which the Kergin % interpolant can be defined. Some (local uniform) convergence results for % Kergin interpolants with increasing numbers of interpolation points are given %AnderssonPassare91b % shayne \refJ Andersson, M., Passare, M.; Complex Kergin interpolation and the Fantappie transform; \MZ; 208(2); 1991; 257--271; % More on complex Kergin interpolation %Bloom79 % shayne \refJ Bloom, T.; Polynomial interpolation; Bol.\ de Soc.\ Bras.\ de Mat.; 10; 1979; xxx--xxx; %Bloom81 % shayne \refJ Bloom, T.; Kergin interpolation of entire functions on $\Cn$; Duke Math.\ J.; 48(1); 1981; 69--83; % local uniform convergence of Kergin interpolants to entire functions as the % number of interpolation points increases %Bloom84 % shayne \refQ Bloom, T.; On the convergence of interpolating polynomials for entire functions; (Analyse Complexe), xxx (ed.), Lecture notes in Math., Vol. 1094, Springer-Verlag (Berlin); 1984; 15--19; % local uniform convergence of Kergin interpolants to entire functions as the % number of interpolation points increases %BloomCalvi94 % shayne \refR Bloom, T., Calvi, J. P.; Kergin interpolants of holomorphic functions; preprint; 1994; %BojanovHakopianSahakian93 % shayne \refB Bojanov, B. D., Hakopian, H. A., Sahakian, A. A.; Spline functions and multivariate interpolations; Kluwer Academic Publishers; 1993; % 14 Chapters % Lagrange, Hermite, Birkhoff interpolation. Budan-Fourier theorem % splines with multiple knots % B-splines, Peano's kernel theorem, Tschakaloff's formula % Hermite, Birkhoff interpolation by splines, Total positivity % interpolation by natural splines, Holladay's theorem % (oscillating) perfect splines, Favard's interpolation problem % monosplines and quadrature formulae % periodic splines % multivariate B-splines and truncated powers % multivariate divided differences % box splines % scale of mean value interpolations, Kergin and Hakopian interpolation % multivariate polynomial interpolation by traces on manifolds % multivariate Birkhoff interpolation from a space spanned by monomials %Bos83a % shayne \refJ Bos, L.; On Kergin interpolation in the disk; \JAT; 37; 1983; 251--261; % max norm error bounds for the Kergin interpolant to points equally spaced % along the boundary of the unit disc %CavarettaGoodmanMicchelliSharma83 % sherm, shayne \refJ Cavaretta, A. S., Micchelli, C. A., Sharma, A.; Multivariate interpolation and the Radon transform, part III: Lagrange representation; Can. Math. Soc. Conf. Proc.; 3; 1983; 37--50; % More on lifting. Studies the choice of a bases for the interpolation % conditions of maps from the family which Hakopian describes as the `scale % of mean value interpolations'. The points of interpolation must satisfy % certain geometric conditions, in which case the dual basis of `Lagrange % polynomials' is given %CavarettaMicchelliSharma80a % larry, shayne \refJ Cavaretta, A. S., Micchelli, C. A., Sharma, A.; Multivariate interpolation and the Radon transform; \MZ; 174; 1980; 263--279; % The first of three papers dealing with the lifting of univariate polynomial % valued projectors to multivariate maps by using the density of plane waves % includes Kergin interpolation which is the `lift' of Hermite interpolation %CavarettaMicchelliSharma80b % shayne \refP Cavaretta, A. S., Micchelli, C. A., Sharma, A.; Multivariate interpolation and the Radon transform, part II: Some further examples; \BonnII; 49--61; % Additional examples of lifts of univariate maps, including Abel-Gontscharoff, % Lidstone, and the area matching interpolation maps %DahmenMicchelli83a % larry, carl, shayne \refJ Dahmen, W. A., Micchelli, C. A.; On the linear independence of multivariate B-splines II: complete configurations; \MC; 41(163); 1983; 143--163; % Amongst other things, there is discussion of the space of interpolation % conditions for the family of lifted maps referred to as the scale of mean % value interpolations, this includes the map of Kergin and Hakopian %DokkenLyche78 % shayne \refR Dokken, T., Lyche, T.; A divided difference formula for the error in numerical differentiation based on Hermite interpolation; Research Report 40, Institute of informatics, Univ. Oslo; 1978; % published as %DokkenLyche79 . See comments there. Kergin %DokkenLyche79 % tom, shayne \refJ Dokken, T., Lyche, T.; A divided difference formula for the error in Hermite interpolation; \BIT; 19; 1979; 540--542; % Gives a formula for the derivatives of the error in Hermite interpolation % which involves only divided differences of order one higher than the degree % of the polynomial space of interpolants. This is in contrast to the formula % that one obtains simply by differentiating the error formula for Hermite % interpolation using the rule for differentiating a divided difference with % respect to one of its knots - which involves additional higher order divided % differences. % The same formula was given at about the same time by %Wang78 %DynLorentzRiemenschneider82 % shayne \refJ Dyn, N., Lorentz, G. G., Riemenschneider, S. D.; Continuity of Birkhoff interpolation; \SJNA; 19(3); 1982; 507--509; % uses `de-coalescence' of the interpolation matrix to show that Birkhoff % interpolation depends continuously on the points of interpolation %Gao88 % shayne \refJ Gao, J. B.; Multivariate quasi-Newton interpolation; J. Math.\ Res.\ Exposition (in Chinese); 8(3); 1988; 447--453; % The third in a series of papers from China (the first two are by Lai and % Wang) that deal with Kergin interpolation and its generalisations. % This paper describes the interpolant and its error for the `scale of mean % value interpolations'. Several complicated formulae involving `multivariate % divided differences' are given. For a description see: Integral error formulae % for the scale of mean value interpolations which includes Kergin and Hakopian % interpolation, by S. Waldron %Goodman83b % sonya, shayne \refJ Goodman, T. N. T.; Interpolation in minimum semi-norm, and multivariate B-splines; \JAT; 37; 1983; 212--223; % The original paper that descibes the family of lifted maps that Hakopian % calls the scale of mean value interpolations, this includes the Kergin map %Hakopian81 % carl, shayne \refJ Hakopian, H.; Les differences divis\'ees de plusieurs variables et les interpolations multidimensionnelles de types Lagrangien et Hermitien; \CRASP\ Ser.\ I; 292; 1981; 453-456; % Description of Hakopian interpolation by using `multivariate divided % differences' %Hakopian82 % carl, shayne \refJ Hakopian, H.; Multivariate divided differences and multivariate interpolation of Lagrange and Hermite type; \JAT; 34; 1982; 286--305; % Description of Hakopian interpolation by using `multivariate divided % differences' %Hollig86c % carl, shayne \refP H\"ollig, K.; Multivariate splines; \Neworleans; 103--127; %HolligMicchelli87 % greg, shayne \refJ H\"ollig, K., Micchelli, C. A.; Divided differences, hyperbolic equations, and lifting distributions; \CA; 3; 1987; 143--156; % More on lifting, multivariate polynomial interpolation and the inverting the % Radon transform %LaiWang84 % carl, shayne \refJ Lai, Mingjun, Wang, Xinghua; A note to the remainder of a multivariate interpolation polynomial; \JATA; 1(1); 1984; 57--63; % Describes the error in Hakopian interpolation using `multivariate divided % differences. For a description see: Integral error formulae for the scale of % mean value interpolations which includes Kergin and Hakopian interpolation, % by S. Waldron. A generalisation Lai and Wang's work is later given by Gao. %LaiWang86 % shayne \refJ Lai, Mingjun, Wang, Xinghua; On multivariate Newtonian interpolation; Sci.\ Sinica\ Ser.\ A; 29(1); 1986; 23--32; % see comments on %LaiWang84 %Liang86 % shayne \refJ Liang, X.; Kergin-interpolation at the points which are zeros of the bivariate polynomial of least deviation from zero on the disk; Northeastern Math.\ J.; 2; 1986; 408-414; % along the same lines as the earlier work of Bos on Kergin interpolation at % points equally spaced along the boundary of the disk %Lorentz92 % shayne \refB Lorentz, R. A.; Multivariate Birkhoff interpolation; Springer-Verlag; 1992; % Birkhoff interpolation from spaces of multivariate polynomials that are % spanned by monomials %Maier94 % shayne \refD Maier, U.; Approximation durch Kergin-Interpolation; dissertation, Universit\"at Dortmund (Germany); 1994; % thesis committee: W. M. M\"uller, M. Reimer, K. Jetter % contains expansions of the standard formula for the interpolant and error % in terms of monomials % Chapters: Grundlagen und Hilfsmittel, Koeffizienten der Kergin- % Interpolierenden, Approximation von Monomen durch Kergin-Interpolierende, % Approximation von Funktionen aus $C^\mu(\RR^n)$ %Wang78 % shayne \refJ Wang, Xinghua; The remainder of numerical differentiation formul{\ae}; Hang Zhou Da Xue Xue Bao (in Chinese); 4(1); 1978; 1--10; % Gives a formula for the derivatives of the error in Hermite interpolation % which involves only divided differences of order one higher than the degree % of the polynomial space of interpolants. This is in contrast to the formula % that one obtains simply by differentiating the error formula for Hermite % interpolation using the rule for differentiating a divided difference with % respect to one of its knots - which involves additional higher order divided % differences. % The same formula was given at about the same time by %DokkenLyche78 %Wang79 % shayne \refJ Wang, Xinghua; On remainders of numerical differentiation formulas; Ke Xue Tong Bao (in Chinese); 24(19); 1979; 869--872; % see comments for %Wang78 multivariate polynomial Hermite interpolation %Here are some added by Jean-Paul Calvi %Bloom90 % shayne \refJ Bloom, T.; Interpolation at discrete subsets of $\CC^n$; \IUMJ; %\jour Indiana Math J. 39(4); 1990; 1223--1243; %Calvi93 % shayne \refJ Calvi, J. P.; Interpolation in Fr\'echet spaces with an application to complex function theory; Indag.\ Math.; 4(1); 1993; 17-26; %Calvi93 % shayne \refJ Calvi, J. P.; Interpolation with prescrived analytic functionals; \JAT; 75(2); 1993; 136-156; %BloomBos83 \refP Bloom, T., Bos, L.; On the convergence of Kergin interpolant of analytic functions; \TexasIV; 369--374; %Calvi93 % shayne \refJ Calvi, J. P.; A convergence problem for Kergin interpolation; Proc.\ Edin.\ Math.\ Soc.; 37; 1994; 175--183; %BloomCalvi95 % shayne \refQ Bloom, T., Calvi, J. P.; A convergence problem for Kergin interpolation II; (Approximation Theory VIII), E. W. Cheney, C. Chui, and L. Schumaker (eds.), Academic Press (New York); 1992; xxx-xxx; %There are also several papers and communications by T. Bloom on a %several variable version (connected with Kergin interpolation) of the %Muntz satz theorem and a "Compte rendu de l'academie des sciences de %Paris" % de Xing Yang related to the same problem. Tom should point out these %references %Tom Bloom provided the following %GoodmanSharma84 % shayne \refJ Goodman, T. N. T., Sharma, A.; Convergence of multivariate polynomials interpolating on a triangular array; \TAMS; 285(1); 1984; 141--157; %Yang91 % shayne \refJ Yang, X.; Une generalisation a plusieurs variables du theoreme de Muntz-Szasz; \CRASP; 312; %Serie I 1991; 575--578; %Bloom90 % shayne \refJ Bloom, T.; A spanning set for ${\cal C}(I^n)$; \TAMS; 34(2); 1990; 740--759; %Bloom90 % shayne \refJ Bloom, T.; A multivariable version of the Muntz-Szasz theorem; Contemporary Math; 137; 1992; 85--92;