%PagalloPereyra1982 % sherm, proceedings update (new one in proceed.tex) \rhl{P} \refP Pagallo, G., Pereyra, V.; Smooth monotone spline interpolation; \Hennart; 142--146; %Pai1982 \rhl{P} \refR Pai, D. V.; On minimization of sublinear functionals and generalized splines; xx; 1982; %PaihiaUtreras1978 \rhl{P} \refJ Paihia, M. L., Utreras, F. I.; Une ensemble de programmes pour l'interpolation des fonctions spline du type plaque mince; Math.\ Appl.\ Informat.\ Rapp.\ Rech.; 140; 1978; 1--61; %Paihua1977 \rhl{P} \refR Paihua, L.; Methodes num\'eriques pour l'obtention de fonctions-spline du type plaque mince en dimension 2; S\'eminaire d'analyse num\'erique \#273, Grenoble; 1977; %Palmer1970 \rhl{P} \refJ Palmer, J. A. B.; An economical method of plotting contours; Austral.\ Computer J.; 2; 1970; 27--31; %Paltanea1988a % shayne 5dec96 \rhl{P} \refJ P\u alt\u anea, R.; Improved estimates with the second order modulus of continuity in approximation by linear positive operators; \ANTA; 17(2); 1988; 171--179; % 91k:41045 %Paltanea2001 % shayne 21jan02 \rhl{} \refQ P{\u a}lt{\u a}nea, R.; On a limit operator; (Proceedings of the Tiberiu Popoviciu Itinerant Seminar of Functional Equations, Approximation and Convexity), E. Popoviciu (ed.), Srima Press (Cluj-Napoca); 2001; 169--179; %PalusznyPatterson \rhl{P} \refR Paluszny, M., Patterson, R. R.; A family of curvature continuous cubic algebraic splines; xx; xx; %PalusznyPatterson1993b % larry \rhl{P} \refJ Paluszny, M., Patterson, R. R.; A family of tangent continuous cubic algebraic splines; \ACMTG; 12; 1993; 209--232; %PalusznyPatterson1994 % larry \rhl{P} \refP Paluszny, M., Patterson, R. R.; $G^2$-continuous cubic algebraic splines and their efficient display; \ChamonixIIa; 353--359; %PalusznyPatterson1999 \rhl{P} \refR Paluszny, M., Patterson, R. R.; Tangent continuous algebraic splines; xxx; xxx; %PalusznyPrautzschSchafer1994 % carlrefs 20nov03 \rhl{} \refR Paluszny, Marco, Prautzsch, Hartmut, Sch\"afer, Martin; Corner cutting and interpolatory refinement; ms, 16dec; 1994; %Pan1988 % carl \rhl{P} \refJ Pan, Wenxi; A generalization of Wiener's theorems concerning the closure of translating span in $L^2(-\infty,\infty)$; \JATA; 4; 1988; 9--11; %Pan1992 % carl \rhl{P} \refJ Pan, Victor; Complexity of computations with matrices and polynomials; \SR; 34; 1992; 225--262; %PanRJ2001 % . 04mar10 \rhl{Q} \refJ Pan, R. J.; Explicit matrix representation for NURBS curves and surfaces and its algorithm; Chinese J.\ Computers; 24(4); 2001; 358--366; %PapamichaelSoares1986 % larry \rhl{P} \refJ Papamichael, N., Soares, M. J.; A posterior corrections for nonperiodic cubic and quintic interpolating splines at equally spaced knots; \IMAJNA; 6; 1986; 489--502; %PapamichaelSoares1987 \rhl{P} \refR Papamichael, N., Soares, M. J.; A class of cubic and quintic spline modified collocation methods for the solution of two-point boundary value problems; Univ.\ do Minho; 1987; %PapamichaelSoares1989 \rhl{P} \refR Papamichael, N., Soares, M. J.; An $O(h^6)$ cubic spline interpolating procedure for harmonic functions; xx; 1989; %PapamichaelSoares1999 \rhl{P} \refR Papamichael, N., Soares, M. J.; Cubic and quintic spline-on-spline interpolation; JCAM; 19xx; %PapamichaelWhiteman1973 % larry \rhl{P} \refJ Papamichael, N., Whiteman, J. R.; A cubic spline technique for the one dimensional heat conduction equation; \JIMA; 11; 1973; 111--113; %PapamichaelWorsey1981 % larry \rhl{P} \refJ Papamichael, N., Worsey, A. J.; End conditions for improved cubic spline derivative approximations; \JCAM; 7; 1981; 101--109; %ParkFerrari2000 % larry 20apr00 \rhl{} \refP Park, Jae H., Ferrari, Leonard A.; The (2-5-2) spline function; \Stmalod; 307--314; %Parker1970 % larry \rhl{P} \refP Parker, J. B.; Methods of graduating hetergenous data; \Hayes; 111--114; %Parker1970b \rhl{P} \refP Parker, J. B.; Experience with cubic splines in the graduation of neutron cross-section data; \Hayes; 107--110; %ParkerDenham1979 % larry \rhl{P} \refJ Parker, R. L., Denham, C. R.; Interpolation of unit vectors; Geophys.\ J. R. Astr.\ Soc.; 58; 1979; 685--687; %ParkerShure1982 % carl 23jun03 \rhl{} \refJ Parker, Robert L., Shure, Loren; Efficient modeling of the earth's magnetic field with harmonic splines; Geophysical Research Letters; 9(8); 1982; 812--815; %Parkinson1992 % carlrefs 08apr04 \rhl{} \refJ Parkinson, D. B.; Optimised biarc curves with tension; \CAGD; 9; 1992; 207--218; %Parr1989 \rhl{P} \refR Parr, W.; Parabolic smoothing and partial least sqare fitting; xxx; xxx; %ParthasarathyElango1989 \rhl{P} \refR Parthasarathy, S., Elango, N.; Least squares cubic spline approximation of empirical data; xxx; xxx; %PasadasTorrensLopezSilanes1997 % larry 10sep99 \rhl{PTL} \refP Pasadas, M., Torrens, J. J., L\'opez de Silanes, M. C.; Approximation of offset curves and surfaces by discrete smoothing $D^m$-splines; \ChamonixIIIa; 329--336; %Paskov1992 % carl \rhl{P} \refJ Paskov, Spassimir H.; Singularity of bivariate interpolation; \JAT; 70; 1992; 50--67; % concerns total degree interpolation to total-degree Hermite data at one or % more points. Choice N:= (n_1,...,n_s, n) of the degrees of the Hermite data % and of the polynomial interpolant is {\bf regular} if there is at least one % choice of points for which this is a correct interpolation problem. If % \sum_j dim\Pi_j < \dim \Pi_n , problem is obviously singular. Guess is that % any singular N is necessarily the (vector) sum of such obviously singular % ones. See also GevorgianHakopianSaakyan92a %Passow1974 % sonya \rhl{P} \refJ Passow, E.; Piecewise montone spline interpolation; \JAT; 12; 1974; 240--241; %Passow1977 % sonya \rhl{P} \refJ Passow, E.; Monotone quadratic spline interpolation; \JAT; 19; 1977; 143--147; %PassowRaymon1975 % larry \rhl{P} \refJ Passow, E., Raymon, L.; The degree of piecewise monotone interpolation; \PAMS; 48; 1975; 409--412; %PassowRoulier1977 % carl \rhl{P} \refJ Passow, Eli, Roulier, John A.; Monotone and convex spline interpolation; \SJNA; 14; 1977; 904--909; %Patent1972 \rhl{P} \refD Patent, P. D.; Least square polynomial spline approximation; Cal.\ Tech.; 1972; %Patent1976 % carl \rhl{P} \refJ Patent, Paul D.; The effect of quadrature errors in the computation of $L_2$ piecewise polynomial approximations; \SJNA; 13; 1976; 344--361; %Patrikalakis1989 % . \rhl{P} \refJ Patrikalakis, Nicholas M.; Approximate conversion of rational splines; \CAGD; 6; 1989; 155--165; %Patterson1978 \rhl{P} \refR Patterson, M. R.; CONTUR: A subroutine to draw contour lines for randomly located data; Oak Ridge; 1978; %Patterson1985 % larry \rhl{P} \refJ Patterson, R. R.; Projective transformations of the parameter of a Bernstein-B\'ezier curve; \ACMTG; 4; 1985; 276--290; %Patterson1988 % larry, carl \rhl{P} \refJ Patterson, R. R.; Parametrizing and graphing nonsingular cubic curves; \CAD; 20(10); 1988; 615--623; % transformation. %Pavalouiu1964 \rhl{P} \refJ Pavalouiu, XX.; Sur l'interpolation a l'aide de polynomes raccordes; Matematica; 6; 1964; 295--299; %Pavlidis1972 \rhl{P} \refQ Pavlidis, T.; Piecewise approximations of functions of two variables through regions with variable boundaries; (Proc.\ ACM Conf.), xxx (ed.), xxx (xxx); 1972; 631--636; %Pavlidis1973 \rhl{P} \refJ Pavlidis, T.; Waveform segmentation through functional approximation; IEEE. Trans.\ Comput.; C22; 1973; 689--697; %Pavlidis1974 \rhl{P} \refR Pavlidis, T.; The use of algorithms of piecewise approximations for picture processing applications; Princeton; 1974; %Pavlidis1975 \rhl{P} \refJ Pavlidis, T.; Optimal piecewise polynomial $L_2$ -approximation of functions of one and two variables; IEEE. Trans.\ Comput.\ C; 24; 1975; 98--102; % free knots %PavlidisHorowitz1973 \rhl{P} \refR Pavlidis, T., Horowitz, S. L.; Segmentation of plane curves; Princeton; 1973; %PavlidisHorowitz1973b \rhl{P} \refQ Pavlidis, T., Horowitz, S.; Piecewise approximation of plane curves; (Proc.\ First Int.\ Joint Conf.\ Pattern Recog.), xxx (ed.), xxx (xxx); 1973; 396--440; %PavlidisMaika1974 % sonya \rhl{P} \refJ Pavlidis, T., Maika, A.; Uniform piecewise polynomial approximation with variable joints; \JAT; 12; 1974; 61--69; % free knots %Pavlov1981 % MR 08apr04 \rhl{} \refJ Pavlov, N. N.; Boundary conditions in the problem of smoothing by cubic splines (Russian); Vychisl.\ Sistemy; 87; 1981; 53--61; % MR84d:41018 %Pavlov1991 \rhl{P} \refR Pavlov, N. N.; The limits of the periodic splines in the convex set as their degree tends to infinity; Traunstein; xxx; %PavlovVershinin1988 % larry \rhl{P} \refJ Pavlov, N. N., Vershinin, V. V.; On the stable approximation of derivatives by splines in the convex set; Mathematica Balkanica; 2; 1988; 222--229; %Payne1970 % carl \rhl{P} \refP Payne, J. A.; An automatic curve-fitting package; \Hayes; 98--106; %PayneToga1999 \rhl{P} \refJ Payne, B. A., Toga, A. W.; Surface reconstruction by multiaxial triangulation; \CGA; xx; xx; xx--xx; %Peach1944 % . \rhl{P} \refJ Peach, M. O.; Simplified technique for constructing orthonormal functions; \BAMS; 50; 1944; 556--641; % Gram-Schmidt %Peano1913a % shayne 26oct95 \rhl{P} \refJ Peano, G.; Resto nelle formule di quadratura espresso con un integralo definito; Atti della reale Acad.\ dei Lincei, Rendiconti (5); 22; 1913; 562--569; % the original paper containing Peano's kernel theorem %Peano1914 % . 26aug98 \rhl{P} \refJ Peano, G.; Residuo in formulas de quadratura; Mathesis (4); 34; 1914; 5--10; %PecaricZwick1999 \rhl{P} \refR Pecaric, J. E., Zwick, D.; $n$-convexity and majorization; Bonn; 19xx; %PegnaWolter1990 \rhl{P} \refJ Pegna, J., Wolter, F.-E.; Geometrical criteria to guarantee curvature continuity of blend surfaces; J. of Mech.\ Design; xxx; submitted; xxx; %PegnaWolter1999a \rhl{P} \refR Pegna, J., Wolter, F.-E.; Designing and mapping trimming curves on surfaces using orthogonal projection; xxx; xxx; %PegnaWolter1999b \rhl{P} \refR Pegna, J., Wolter, F.-E.; A simple criterion to guarantee second order smoothness of blend surfaces; xxx; xxx; %PelosiFaroukiManniSestini2005 % carl 03apr06 \rhl{} \refJ Pelosi, Francesca, Farouki, Rida T., Manni, Carla, Sestini, Alessandra; Geometric Hermite interpolation by spatial Pythagorean-hodograph cubics; \AiCM; 22(4); 2005; 325--352; %PeltoElkinsBoyd1968 \rhl{P} \refJ Pelto, C., Elkins, T., Boyd, H.; Automatic contouring of irregularly spaced data; Geophysics; 33; 1968; 424--430; %Peluso1975 \rhl{P} \refJ Peluso, I. R.; Un problema di interpolazione con funzioni spline; Bolletino U. M. I.; 11; 1975; 240--251; %Pena2000 % larry 20apr00 \rhl{} \refP Pe{\~n}a, J. M.; Error analysis of algorithms for evaluating Bernstein-B\'ezier-type multivariate polynomials; \Stmalod; 315--324; %PenaSauer0x % author \rhl{P} \refJ Pe{\~n}a, J. M., Sauer, T.; Efficient polynomial reduction; \AiCM; xx; 200x; xxx--xxx; %PenaSauer2000a % . 14may99 21jan02 \rhl{P} \refJ Pe{\~n}a, J. M., Sauer, T.; On the multivariate Horner scheme; \SJNA; 37; 2000; 1186--1197; %PenaSauer2000b % author 21jan02 \rhl{P} \refJ Pe{\~n}a, J. M., Sauer, T.; On the multivariate Horner scheme II: Running error analysis; \C; 65; 2000; 313--322; %Pence1976 % larry \rhl{P} \refD Pence, D. D.; Best approximation by constrained splines; Purdue Univ.; 1976; %Pence1979 % sonya \rhl{P} \refJ Pence, D. D.; Hermite-Birkhoff interpolation and monotone approximation by splines; \JAT; 25; 1979; 248--257; %Pence1980 % larry \rhl{P} \refP Pence, D. D.; Computing nonlinear spline functions; \TexasIII; 721--722; %Pence1980b % sonya \rhl{P} \refJ Pence, D. D.; Best mean approximation by splines satisfying generalized convexity constraints; \JAT; 28; 1980; 333--348; %Pence1983 % larry \rhl{P} \refP Pence, D. D.; Asymptotic properties of approximation by splines with multiple knots; \TexasIV; 653--656; %PenceSmith1982 % larry 5dec96 \rhl{P} \refJ Pence, D. D., Smith, P. W.; Asymptotic properties of best $L_p[0,1]$ approximation by splines; \SJMA; 13; 1982; 409--420; %PengWaldron2000 % shayne 16aug02 \rhl{} \refJ Peng, I., Waldron, S.; Signed frames and Hadamard products of Gram matrices; \LAA; 347(1--3); 2000; 131--157; %PennbakerMitchellLandon1988a \rhl{P} \refJ Pennbaker, W. B., Mitchell, J. L., Landon Jr., G. G.; An overview of the principles of the Q-coder adaptive binary arithmetic coder; \IBMJRD; 32; 1988; 717--727; %Percell1976 % carl \rhl{P} \refJ Percell, Peter; On cubic and quartic Clough--Tocher elements; \SJNA; 13; 1976; 100--103; %Pereverzev1981 \rhl{P} \refJ Pereverzev, S. V.; Sharp estimates of approximation by Hermitian splines on a class of differentiable functions of two variables (Russian); Izv.\ Vyssh.\ Ucebn.\ Zaved.\ Math.; 12; 1981; 58--66; %PerezPinar1996 % shayne 26aug98 \rhl{P} \refJ P\'erez, T. E., Pi\~nar, M. A.; On Sobolev orthogonality for the generalized Laguerre polynomials; \JAT; 86(3); 1996; 278--285; %PerrinPriceVarga1969 % larry \rhl{P} \refJ Perrin, F., Price, H., Varga, R. S.; On higher-order numerical methods for nonlinear two-point boundary value problems; \NM; 13; 1969; 180--198; %Persson1978a \rhl{P} \refJ Persson, H.; NC machining of arbitrarily shaped pockets; Computer Aided Design; 10; 1978; 170--174; %PescoTavares1994 % larry \rhl{P} \refP Pesco, S., Tavares, G.; Splines in a topological setting; \ChamonixIIa; 361--368; %Pesenson1995 % author 21jan02 \rhl{} \refR Pesenson, I.; Lagrangian splines, spectral entire functions and Shannon-Whittaker theorem on manifolds; Resarch Report 95-87 (28pp), Temple Univ.{} (???); 1995; %PeternellPottmann2000 % larry 20apr00 \rhl{} \refP Peternell, Martin, Pottmann, Helmut; Interpolating functions on lines in 3-space; \Stmalof; 351--358; %Peters1974 \rhl{P} \refP Peters, G. J.; Interactive computer graphics application of the parametric bicubic surface to engineering design problems; \Barnhill; 259--302; %Peters1988a \rhl{P} \refR Peters, G. J.; Local piecewise cubic $C^1$ surface interpolants for rectangular and triangular tessellations; CMS 89-10; 1988; %Peters1988b \rhl{P} \refR Peters, G. J.; Local piecewise cubic $C^1$ surface interpolants via splitting and averaging; CMS 89-11; 1988; %Peters1988c \rhl{P} \refR Peters, G. J.; Generalized Hermite interpolation by quartic $C^2$ space curves; CMS 89-12; 1988; %Peters1989a \rhl{P} \refJ Peters, J.; Local generalized Hermite interpolation by quartic $C^2$ space curves; \ACMTG; 8(2); 1989; 235--242; %Peters1989b % author \rhl{P} \refR Peters, J.; Rectangulation Algorithms: Smooth Surface Interpolation with Bicubics; CMS Tech.\ Report No.\ 90-1, U.Wisconsin-Madison; 1989; %Peters1990a % carl \rhl{P} \refJ Peters, J.; Smooth mesh interpolation with cubic patches; \CAD; 22(2); 1990; 109--120; % geometric design, Bernstein-B\'ezier form, surfaces. %Peters1990b % greg \rhl{P} \refJ Peters, J.; Local smooth surface interpolation: a classification; \CAGD; 7; 1990; 191--195; %Peters1990c % greg \rhl{P} \refJ Peters, J.; Local cubic and bicubic $C^1$ surface interpolation with linearly varying boundary normal; \CAGD; 7; 1990; 499--516; %Peters1990d % author \rhl{P} \refD Peters, J.; Fitting smooth parametric surfaces to 3D data; University of Wisconsin, CMS Tech.\ Report No.\ 91-2; 1990; %Peters1991a % carl \rhl{P} \refJ Peters, J.; Smooth interpolation of a mesh of curves; \CA; 7; 1991; 221--246; %Peters1991b % author \rhl{P} \refQ Peters, J.; Parametrizing singularly to enclose vertices by a smooth parametric surface; (Proceedings of Graphics Interface '91), S. MacKay, E. M. Kidd (eds.), Canadian Man-Computer Communications Society (xxx); 1991; 1--7; %Peters1992a % greg \rhl{P} \refJ Peters, J.; Joining smooth patches around a vertex to form a $C^k$ surface; \CAGD; 9; 1992; 387--411; %Peters1992b % author \rhl{P} \refQ Peters, J.; Improving $G^1$ surface joins by using a composite patch; % Proceedings, SPIE Conference , Nov 16--18 1992, (Curves and Surfaces in Computer Vision and Graphics III), J. D. Warren (ed.), xxx (xxx); 1992; 345--354; %Peters1993a % author \rhl{P} \refR Peters, J.; Smooth splines over irregular meshes built from few polynomial pieces of low degree; CSD-TR-93-019 Purdue University; 1993; %Peters1993b % greg \rhl{P} \refJ Peters, J.; Smooth free-form surfaces over irregular meshes generalizing quadratic splines; \CAGD; 10; 1993; 347--361; % CSD-TR-92-063, Purdue U.: 1993: %Peters1994a % carl \rhl{P} \refR Peters, Jorg; Curvature continuous spline surfaces over irregular meshes; ms, Comp.Sci.\ Purdue U.; 1994; %Peters1994b % author \rhl{P} \refQ Peters, J.; Surfaces of arbitrary topology constructed from biquadratics and bicubics; (Designing fair curves and surfaces), N. Sapidis (ed.), SIAM Publications (xxx); 1994; xxx-xxx; %Peters1994c % author \rhl{P} \refJ Peters, J.; Smoothing vertex-degree bounded polyhedra; \ACMTG; xx; 199x; xxx--xxx; %Peters1994d % author \rhl{P} \refJ Peters, J.; Evaluation of multivariate Bernstein polynomials; \ACMTG; xx; 199x; xxx--xxx; %Peters1994f % larry \rhl{P} \refP Peters, J.; A characterization of connecting maps as nonlinear roots of the identity; \ChamonixIIa; 369--376; %Peters1994g % carl 5dec96 \rhl{P} \refR Peters, J.; Biquartic C1 spline surfaces over irregular meshes; CSD-TR-94-059, Computer Sciences Department, Purdue University (West Lafayette, IN), August; 1994; %Peters1994h % carl 5dec96 \rhl{P} \refR Peters, J.; Interpolation regions for convex low degree polynomial curve segments; CSD-TR-94-064, Computer Sciences Department, Purdue University (West Lafayette, IN), September; 1994; % sharp conditions on the location of an intermediate point to ensure that a % parametric quadratic or cubic curve matching given end conditions can pass % through that point, yet have no inflection point. %Peters1995 % author 22may98 \rhl{P} \refJ Peters, J.; $C^1$ free-form surface splines; \SJNA; 32(2); 1995; 645--666; %Peters1996 % peters 05mar08 \rhl{P} \refQ Peters, J\"org; $C^2$ surfaces built from zero sets of the 7-direction box spline; (Mathematics of Surfaces VI {(IMA conf. at Brunel 1994)}), Glen Mullineux (ed.), Clarendon Press, Oxford (England); 1996; 463--474; % ISBN 0-19-851198-1, c %PetersReif1997 % carl 20apr00 \rhl{P} \refJ Peters, J\"org, Reif, Ulrich; The simplest subdivision scheme for smoothing polyhedra; \ACMTG; 16(4); 1997; 420--431; %PetersReif1998 % carl 22may98 \rhl{P} \refJ Peters, J\"org, Reif, Ulrich; Analysis of algorithms generalizing B-spline subdivision; \SJNA; 35(2); 1998; 728--748; % bivariate (biquadratic and bicubic), smoothness of limiting surface %PetersReif2000 % carl 02feb01 \rhl{P} \refJ Peters, J\"org, Reif, Ulrich; Least squares approximation of B\'ezier coefficients provides best degree reduction in the $L_2$-norm; \JAT; 104(1); 2000; 90--97; %PetersReif2005 % author \rhl{Q} \refQ Peters, J\"org, Reif, Ulrich; Topics in multivariate approximation and interpolation; (Structural Analysis of Subdivision Surfaces - A Summary), K. Jetter et al. (eds.), xxx (xxx); 2005; 149--190; %PetersSitharam1990 \rhl{P} \refR Peters, J., Sitharam, M.; Interpolation from $C^1$ cubics at the vertices of an underlying triangulation; Univ.\ of Wisconsin-Madison; 1990; %PetersSitharam1992a % sherm, pagination update \rhl{P} \refJ Peters, J., Sitharam, M.; Stability of interpolation from $C^1$ cubics at the vertices of an underlying triangulation; \SJNA; 29(2); 1992; 528--533; %PetersSitharam1992b % author \rhl{P} \refJ Peters, J., Sitharam, M.; On stability of $m$-variate $C^1$ interpolation; \JATA; 8; 1992; 17--32; %PetersWittman1997 % peters 05mar08 \rhl{PW} \refJ Peters, J., Wittman, M.; Box-spline based CSG blends; Proc.\ fourth ACM symposium on Solid modeling and applications, SIGGRAPH, ACM Press; ; 1997; 195--205; %Petersen1983 \rhl{P} \refR Petersen, C. S.; Contours of three and four-dimensional surfaces; M.S. Thesis, Univ.\ Utah; 1983; %Petersen1984 % greg \rhl{P} \refJ Petersen, C. S.; Adaptive contouring of three dimensional surfaces; \CAGD; 1; 1984; 61--74; %PetersenI1962a % . 14sep95 \rhl{P} \refJ Petersen, I.; Piecewise-polynomial approximation; Izv.\ AN Est.SSR, Ser.\ Fiz.-Matem.\ i Tekhn.\ Nauk; 1; 1962; xxx--xxx; % taken from %Subbotin67b who describes it as containing error bounds, for % function, first and second derivative, for cubic spline interpolation in % the cardinal case. %PetersonCPiperWorsey1987 \rhl{P} \refP Peterson, C. S., Piper, B., Worsey, A. J.; Adaptive contouring of a trivariate interpolant; \Troy; 385--396; %PetersonI1962 \rhl{P} \refJ Peterson, I.; On a piecewise polynomial approximation; Eesti NSV Tead.\ Akad.\ Toimetised Fuus.\ -Mat.; 11; 1962; 24--32; %Petersson2002 % author 21nov08 \rhl{P} \refJ Petersson, H.; Kergin interpolation in Banach spaces; \SM; 153; 2002; 101--114; %Petersson2004 % author 21nov08 \rhl{P} \refJ Petersson, H.; Interpolation spaces for PDE-preserving projectors on Hilbert-Schmidt entire functions; \RMJM; 34(3); 2004; 1059--1075; % Kergin interpolation in Banach spaces %Petit1971 \rhl{P} \refJ Petit, M.; Une propriete des fonctions spline d'ajustement; Rev.\ Francaise Informat.\ Recherche Operationnelle; 5; 1971; 137--140; %Petras1990 %carl 02feb01 \rhl{} \refJ Petras, Knut; On the minimal norms of polynomial projections; \JAT; 62; 1990; 206--212; % sharp asymptotic behavior as $k\to\infty$ of the norm of minimal projector % onto $\Pi_k$. %Petrov1996 % carl 6aug96 \rhl{P} \refJ Petrov, Petar P.; Shape preserving approximation by free knot splines; \EJA; 2(1); 1996; 41--48; % For given p, every monotone or convex function has a ``near best'' free % knot spline approximant which is monotone or convex, of order k with % O(k^2 n) simple knots, and with error, in the L_p-norm, no larger than some % const_p times the error in the (unconstrained) ba to the function by % splines of order k with n free knots. Similar (stronger?) results in % LeviatanShadrin in \ChamonixIII. %Petrushev1987a \rhl{P} \refJ Petrushev, Pencho P.; Relations between rational and spline approximations in $L_p$ metric; \JAT; 50(2); 1987; 141--159; %Petrushev1988 % jia \rhl{P} \refQ Petrushev, P.; Direct and converse theorems for spline and rational approximation and Besov spaces; (Function Spaces and Applications), M. Cwikel, J. Peetre, Y. Sagher and H. Wallin (eds.), Lecture Notes in Math.\ Vol.\ 302, Springer (New York); 1988; 363--377; %PetrushevPopov1987 % carl 03dec99 \rhl{P} \refB Petrushev, P. P., Popov, V. A.; Rational Approximation of Real Functions; Cambridge University Press (Cambridge); 1987; % general approximation theory with special emphasis on rational % approximation, but also spline approximation, esp.\ approximation order of % free knot splines characterized in terms of Besov spaces. %PeuckerFowlerLittleMark1970 \rhl{P} \refR Peucker, T. K., Fowler, R. J., Little, J. J., Mark, D. M.; Digital representation of three-dimensional surfaces by triangulated irregular networks; Rpt.\ 10, Simon Fraser Univ.; 1970; %Pfeifer1988 % carl \rhl{P} \refJ Pfeifer, E.; Interpolation with exponentially fitted second-order $C^1$-spline functions; \JCAM; 21(1); 1988; 119--124; %PfeifleBartelsGoldman1989a \rhl{P} \refP Pfeifle, R., Bartels, R., Goldman, R.; Tensor product slices; \Biri; 431--440; %PfeifleSeidel1995 % LLS Lai-Schumaker book \rhl{PfeS95} \refQ Pfeifle, R., Seidel, H.-P.; Spherical triangular B-splines with application to data fitting; (Computer Graphics Forum, Vol. 14), F. Post and M. G\"obel (eds.), Blackwell (London); 1995; 89--96; %PflugerGmelig1988 \rhl{P} \refR Pfluger, P., Meyling, R. H. J. Gmelig; An algorithm for smooth interpolation to scattered data in $\RR^2$; Twente; 1988; %PflugerNeamtu1991 % carl 26aug98 \rhl{P} \refP Pfluger, P. R., Neamtu, M.; Geometrically smooth interpolation by triangular Bernstein-B\'ezier patches with coalescent control points; \ChamonixI; 363--366; %PflugerNeamtu1993 % carl 26aug98 \rhl{P} \refJ Pfluger, P. R., Neamtu, M.; On degenerate surface patches; \NA; 5; 1993; 569--575; %Pham1988a \rhl{P} \refJ Pham, B.; Offset Approximation of Uniform B-splines; Computer Aided Design; 20; 1988; 471--474; %Phillips1968a % larry \rhl{P} \refJ Phillips, G. M.; Algorithms for piecewise straight line approximations; \CJ; 11; 1968; 211--212; %Phillips1968c % larry \rhl{P} \refJ Phillips, G. M.; Estimate of the maximum error in best polynomial approximations; \CJ; 11; 1968; 110--111; %Phillips1970a % shayne 26oct95 \rhl{P} \refP Phillips, G. M.; Error estimates for best polynomial approximations; \Talbot; 1--6; % bounding the error in the best L_p-approximation to f from polynomials of % degree < n by the max norm of D^n(f). The result follows from my basic % estimate (see ... ) %Phillips1971a % shayne 14sep95 \rhl{P} \refQ Phillips, G.; Error estimates for certain integration rules on the triangle; (Applications of Numerical Analysis), J. Morris (ed.), Springer-Verlag (Berlin); 1971; 321--326; % representation of the Lagrange polynomials for interpolation from % quadratics % to the simplex points in terms of the barycentric coordinates and the % corresponding cubature rule (with error estimate) %Phillips1979 % larry \rhl{P} \refJ Phillips, G. M.; Best polynomial approximations: a corrected proof; \BIT; 19; 1979; 95--97; %Phillips1992a % carl \rhl{P} \refP Phillips, G. M.; Error estimates for near-minimax approximations; \SinghII; 223--241; % rehash of 68c %PhillipsJ1972 % larry \rhl{P} \refJ Phillips, J. L.; The use of collocation as a projection method for solving linear operator equations; \SJNA; 9; 1972; 14--28; %PhillipsJHanson1973 \rhl{P} \refR Phillips, J. L., Hanson, R. J.; Computing integrals involving B-splines by means of specialized quadrature rules; Washington State Univ.; 1973; %PhillipsTaylor1982a % shayne 26oct95 \rhl{P} \refJ Phillips, G. M., Taylor, P. J.; Polynomial approximation using equioscillation on the extreme points of Chebyshev polynomials; \JAT; 36; 1982; 257--264; % shows that if p is a polynomial of degree \le k chosen so that f-p % equioscillates on the point set consisting of the extrema of the Chebyshev % polynomial T_{k+1}, then max norm of the error f-p can be bounded by the % max norm of D^{k+1}f multiplied by the same constant used when bounding the % error in best approximation from \Pi_k in the same way %PhillipsWatsonDF1982 % . 03dec99 \rhl{P} \refJ Phillips, G. M., Watson, D. F.; A precise method for determining contoured surfaces; Austral.{} Petro.{} Expl.{} Assoc.{} J.; 22; 1982; 205--212; %PhillipsWatsonDF1986a % . 03dec99 \rhl{P} \refJ Phillips, G. M., Watson, D. F.; Matheronian Geostatistics---Quo Vadis?; Math.\ Geol.; 18; 1986; 93--117; %PhillipsWatsonDF1986b % . 03dec99 \rhl{P} \refJ Phillips, G. M., Watson, D. F.; Geostatistics and spatial data analysis; Math.{} Geol.; 18; 1986; 505--509; %Pickrell1979 % larry \rhl{P} \refR Pickrell, A. J.; Representation of hydrographic surveys and ocean bottom topography by analytical models; M. S. Thesis, Naval Postgraduate School; 1979; %Piegl1986 % carlrefs 08apr04 \rhl{} \refJ Piegl, L.; Curve fitting algorithm for rough cutting; \CAD; 18(2); 1986; 79--82; %Piegl1987a % carl \rhl{P} \refJ Piegl, Les; Infinite control points - a method of representing surfaces of revolution using boundary data; \ICGA; 7(3); 1987; 45--55; %Piegl1987b % greg \rhl{P} \refJ Piegl, L.; On the use of infinite control points in CAGD; \CAGD; 4; 1987; 155--166; %Piegl1987c % carl \rhl{P} \refJ Piegl, Les; Less data for shapes; \ICGA; 7(8); 1987; 48--49; %Piegl1987d % carl \rhl{P} \refJ Piegl, Les; Interactive data interpolation by rational B\'ezier curves; \ICGA; 7(4); 1987; 45--58; %Piegl1988a % carl \rhl{P} \refJ Piegl, Les; Hermite-- and Coons--like interpolants using rational B\'ezier approximation form with infinite control points; \CAD; 20(1); 1988; 2--10; % geometry. %Piegl1991a % carl \rhl{P} \refJ Piegl, Les; On NURBS, A survey; \ICGA; 11(1); 1991; 55--71; %PieglTiller1987a % carl \rhl{P} \refJ Piegl, Les, Tiller, Wayne; Curve and surface constructions using rational B-splines; \CAD; 19(9); 1987; 485--498; % computer-aided design, geometric modelling. %PieglTiller1994 % carl \rhl{P} \refJ Piegl, Les, Tiller, Wayne; Software-engineering approach to degree elevation of B-spline curves; \CAD; 26(1); 1994; 17--28; % converts to BB-form, as it is `easier to understand', elevates degree, % then recovers B-form, by removing knots. %PieglTiller1995 % . 03dec99 \rhl{P} \refB Piegl, Les, Tiller, Wayne; The NURBS Book; Springer-Verlag (Heidelberg); 1995; % computer-aided design, geometric modelling. %PieglTiller1997 % larry 10sep99 \rhl{PT} \refP Piegl, L., Tiller, W.; Algorithm for computing the product of two B-splines; \ChamonixIIIa; 337--344; %PierceVarga1972a % larry \rhl{P} \refJ Pierce, J. G., Varga, R. S.; Higher order convergence results for the Rayleigh-Ritz method applied to eigenvalue problems: I. Estimates relating Rayleigh-Ritz and Galerkin approximations to eigenfunctions; \SJNA; 9; 1972; 137--151; %PierceVarga1972b % larry \rhl{P} \refJ Pierce, J. G., Varga, R. S.; Higher order convergence results for the Rayleigh-Ritz method applied to eigenvalue problems: 2. Improved error bounds for eigenfunctions; \NM; 19; 1972; 155--169; %Pieroni1968 \rhl{P} \refJ Pieroni, G.; An interpolation method for the automatic drawing of a plane curve; Calcolo; 5; 1968; 173--180; %Pietz1999 \rhl{P} \refR Pietz, K.; $L_p$ approximation by Chebyshevian spline functions; xx; 19xx; %PigounakisKaklis1997 % larry 10sep99 \rhl{PK} \refP Pigounakis, K. G., Kaklis, P. D.; Smoothing spatial cubic B-splines under shape constraints; \ChamonixIIIa; 345--354; %Pilcher1974 \rhl{P} \refP Pilcher, D. T.; Smooth parametric surfaces; \Barnhill; 237--254; %Pinkus1974 % author \rhl{P} \refJ Pinkus, A.; Representation theorems for tchebycheffian polynomials with boundary conditions and their applications; \IsJM; 17; 1974; 11--34; %Pinkus1975 \rhl{P} \refJ Pinkus, A.; Asymptotic minimum norm quadrature formulae; % thesis, Weizmann Inst., 1975? \NM; 24; 1975; 163--175; %Pinkus1976a % larry \rhl{P} \refJa Pinkus, A.; Best $L^1$ approximation by spline functions with fixed knots; \JAT; 18; 1976; 130--135; %Pinkus1976b % sonya \rhl{P} \refJ Pinkus, A.; One-sided $L^1$ approximation by spline functions with fixed knots; \JAT; 18; 1976; 130--135; %Pinkus1976c % author \rhl{P} \refJ Pinkus, A.; A simple proof of the Hobby-Rice theorem; \PAMS; 60; 1976; 82--84; %Pinkus1976d % author \rhl{P} \refP Pinkus, A.; Applications of representation theorems to problems of Chebyshev approximation with constraints; \Karlin; 83--111; %Pinkus1978 \rhl{P} \refJ Pinkus, A.; Some extremal properties of perfect splines and the pointwise Landau problem on the finite interval; % MRC 1678; 1976; \JAT; 23; 1978; 37--64; %Pinkus1979a % author \rhl{P} \refJ Pinkus, A.; Matrices and $n$-widths; \LAA; 27; 1979; 245--278; %Pinkus1979b % author \rhl{P} \refJ Pinkus, A.; On $n$-widths of periodic functions; \JAM; 35; 1979; 209--235; %Pinkus1981a % sonya \rhl{P} \refJ Pinkus, A.; Best approximations by smooth functions; \JAT; 33; 1981; 147--178; %Pinkus1981b % author 02feb01 \rhl{P} \refJ Pinkus, A.; Bernstein's comparison theorem and a problem of Braess; \AeM; 23; 1981; 98--107; %Pinkus1983 % author \rhl{P} \refP Pinkus, A.; $n$-widths in Approximation Theory: A survey; \TexasIV; 153--186; %Pinkus1985a % carl \rhl{P} \refJ Pinkus, A.; $n$-widths of Sobolev spaces in $L^p$; \CA; 1; 1985; 15--62; %Pinkus1985b % carl \rhl{P} \refB Pinkus, A.; $n$-Widths in Approximation Theory; Ergebnisse der Math.\ 3. Folge, Band 7, x+291pp, Springer (Heidelberg); 1985; %Pinkus1985c % author \rhl{P} \refJ Pinkus, A.; Some extremal problems for strictly totally positive matrices; \LAA; 64; 1985; 141--156; %Pinkus1986a % carl \rhl{P} \refP Pinkus, A.; $N$-widths and optimal recovery; \Neworleans; 51--66; %Pinkus1986b % author \rhl{P} \refJ Pinkus, A.; Unicity subspaces in $L^1$-approximation; \JAT; 48; 1986; 226--250; %Pinkus1988a % author \rhl{P} \refJ Pinkus, A.; On smoothest interpolants; \SJMA; 19; 1988; 1431--1441; %Pinkus1988b % author \rhl{P} \refJ Pinkus, A.; Continuous selections for the metric projection on $C_1$; \CA; 4; 1988; 85--96; %Pinkus1989a % carl \rhl{P} \refB Pinkus, A.; On $L^1$-Approximation; Cambridge Tracts in Mathematics 93, x+239pp, Cambridge University Press (xxx); 1989; %Pinkus1993a % carl \rhl{P} \refJ Pinkus, A.; Uniqueness in vector-valued approximation; \JAT; 73(1); 1993; 17--92; %Pinkus1995 % carl 07may96 \rhl{P} \refP Pinkus, A.; Some density problems in multivariate approximation; \IDoMATnifi; 277--284; %Pinkus1996 % shayne 22may98 \rhl{P} \refQ Pinkus, A.; Spectral properties of totally positive kernels and matrices; (Total positivity and its applications), M. Gasca, C. A. Micchelli (eds.), Kluwer Acad.\ Publ.{} (Dordrecht); 1996; 477--511; %Pinkus1997 % larry 10sep99 \rhl{P} \rhl{P} \refP Pinkus, A.; Approximating by ridge functions; \ChamonixIIIb; 279--292; %Pinkus1999 % carl 26aug99 \rhl{P} \refJ Pinkus, Allan; Approximation theory of the MLP model in neural networks; \AN; 8; 1999; 143--196; %Pinkus1999b \rhl{P} \refR Pinkus, A.; Unicity subspaces in $L_1$ approximation; Technion; xx; %Pinkus2000 % carl 20apr00 \rhl{P} \refJ Pinkus, Allan; On a problem of G. G. Lorentz; \JAT; 103; 2000; 29--54; % characterize uniqueness sets of the form U/V := {u/v: u in U, v in V} in % C(B), with U, V finite-dim lss. %PinkusKarlin1976 \rhl{P} \refR Pinkus, A., Karlin, S.; Divided differences and other nonlinear existence problems at extremal points; IBM; 1976; %PinkusShisha1981 % author \rhl{P} \refQ Pinkus, A., Shisha, O.; A variation on the Chebyshev theory of best approximation; (Constructive Function Theory, 81), xxx (ed.), xxx (Sofia); 1983; 479--481; %PinkusShisha1982 % author \rhl{P} \refJ Pinkus, A., Shisha, O.; Variations on the Chebyshev and $l^q$ theories of best approximation; \JAT; 35; 1982; 148--168; %PinkusStrauss1987 \rhl{P} \refJ Pinkus, A., Strauss, H.; One-sided $L^1$-approximation to differentiable functions; \ATA; 3; 1987; 81--96; %PinkusStrauss1988 % author \rhl{P} \refJ Pinkus, A., Strauss, H.; Best approximation with coefficient constraints; \IMAJNA; 8; 1988; 1--22; %PinkusStrauss1990 % author \rhl{P} \refJ Pinkus, A., Strauss, H.; $L^1$-approximation with constraints; \TAMS; 322; 1990; 239--261; %PinkusTotik1986 % author \rhl{P} \refJ Pinkus, A., Totik, V.; One-sided $L^1$-approximation; \CMB; 29; 1986; 84--90; %PinkusWajnryb1988 % author \rhl{P} \refJ Pinkus, A., Wajnryb, B.; Necessary conditions for uniqueness in $L^1$-approximation; \JAT; 53; 1988; 54--66; %PinkusWajnryb1995a % carl 14sep95 \rhl{P} \refJ Pinkus, A., Wajnryb, B.; Multivariate polynomials: a spanning question; \CA; 11(2); 1995; 165--180; % ridge functions, plane waves. With $g$ a polynomial, and $d=2$, % $((g(\cdot-a))^k: a\in \Rd, k=0,1,2,\ldots)$ is dense in $\Pi$ iff % $(g(\cdot-a): a\in \Rd)$ separates points. Not true when $d>3$. %PinkusWulbert1992 % author \rhl{P} \refJ Pinkus, A., Wulbert, D.; The multi-dimensional von Neumann alternating direction search algorithm in $C(B)$ and $L_1$; \JFA; 104; 1992; 121--148; %PinkusZiegler1979 % author \rhl{P} \refJ Pinkus, A., Ziegler, Z.; Interlacing properties of the zeros of the error functions in best $L^p$ approximation; \JAT; 27; 1979; 1--18; %Piper1987a \rhl{P} \refP Piper, B. R.; Visually smooth interpolation with triangular B\'ezier patches; \Troy; 221--233; %Piper1987b \rhl{P} \refR Piper, B.; Continuous triangulations; CAGD; 1987; %Piper1992 \rhl{P} \refR Piper, B.; Properties of local coordinates based on Dirichlet tesselations; Rensselaer Polytechnic Institute, Troy, New York; 1992; %Piper1993 % . 20nov03 \rhl{} \refP Piper, B.; Properties of local coordinates based on Dirichlet tessellations; \Farinnith; 227--240; %Piper1999 \rhl{P} \refR Piper, B.; Review of two dimensional triangulations; Utah; xx; %PivovarovaPuchnacheva1975 \rhl{P} \refR Pivovarova, N. B., Puchnacheva, P.; Smoothing experimental data with local splines; Novosibirsk; 1975; %Plaskota1993 % carl \rhl{P} \refJ Plaskota, L.; Optimal approximation of linear operators based on noisy data on functionals; \JAT; 73(1); 1993; 93--105; %PlassStone1983 \rhl{P} \refJ Plass, Michael, Stone, Maureen; Curve-Fitting with Piecewise Parametric Cubics; Computer Graphics; 17; 1983; 229--239; %Plonka1993a % carl \rhl{P} \refJ Plonka, Gerlind; An efficient algorithm for periodic Hermite spline interpolation with shifted nodes; \NA; 5; 1993; 51--62; %Plonka1994a % carl \rhl{P} \refJ Plonka, Gerlind; Periodic spline interpolation with shifted nodes; \JAT; 76(1); 1994; 1--20; %Plonka1994b % . \rhl{P} \refJ Plonka, G.; Optimal shift parameters for periodic spline interpolation; \NA; 6; 1994; 297--316; %Plonka1994c % larry \rhl{P} \refP Plonka, G.; Spline wavelets with higher defect; \ChamonixIIb; 387--398; %Plonka1995a % author \rhl{P} \refJ Plonka, G.; Two-scale symbol and autocorrelation symbol for B-splines with multiple knots; \AiCM; 3; 1995; 1--22; %Plonka1996 % carl 6aug96 \rhl{P} \refJ Plonka, G.; Generalized spline wavelets; \CA; 12(1); 1996; 127--155; %Plonka1997 % 19nov95, 20jul96, carl 29apr97 \rhl{P} \refJ Plonka, Gerlind; Approximation order provided by refinable function vectors; \CA; 13(2); 1997; 221--244; %Plonka1997a % larry 10sep99 \rhl{P} \rhl{P} \refP Plonka, G.; On stability of scaling vectors; \ChamonixIIIb; 293--300; %PlonkaRon2001 % . 26aug98 20apr00 carl 16mar01 \rhl{PR} \refJ Plonka, G., Ron, A.; A new factorization technique of the matrix mask of univariate refinable functions; \NM; 87(3); 2001; 555--595; %PlonkaTasche1992 % carl 20nov03 \rhl{P} \refJ Plonka, G., Tasche, M.; Efficient algorithms for the periodic Hermite-spline interpolation; \MC; 58(198); 1992; 693--703; % Univ.\ Rostock: 1990: %PocchiolaVegter2000 % larry 20apr00 \rhl{} \refP Pocchiola, Michel, Vegter, Gert; A basis for homogeneous polynomial solutions to homogeneous constant coefficient PDE's: an algorithmic approach through apolarity; \Stmalod; 325--334; %PodolskyDenman1999 \rhl{P} \refJ Podolsky, B., Denman, H. H.; Conditions on minimization criteria for smoothing; xx; xx; xx; xx; %Poeppelmeier1975 % larry \rhl{P} \refR Poeppelmeier, C. C.; A Boolean sum interpolation scheme to random data for computer aided geometric design; M.S. Thesis, Univ.\ Utah; 1975; %Poeschl1984 % greg \rhl{P} \refJ Poeschl, T.; Detecting surface irregularities using isophotes; \CAGD; 1; 1984; 163--168; %Poljak1971 % larry \rhl{P} \refJ Poljak, R. A.; On best convex Chebyshev approximation; Soviet Math.\ Dokl; 12; 1971; 1441--1444; %Polskii1999 \rhl{P} \refR Polskii, N. I.; Projective methods in applied mathematics; xx; 19xx; %Polya1931 % larry carlrefs \rhl{P} \refJ P\'olya, G.; Bemerkung zur Interpolation und zur N\"aherungstheorie der Balkenbiegung; \ZAMM; 11(6); 1931; 445--449; % Polya's sufficient conditions for correctness of Birkhoff interpolation: % The map $f \mapsto (D^{j_k}f(x_k): k=1,\ldots,n)$ with $0\le j_1\le\cdots\le % j_n< n$ and $j_k\not=j_i$ if $x_k=x_i$ is 1-1 on $\Pi_{ knot insertion) %Popoviciu1934b % carl 16aug02 \rhl{P34} \refJ Popoviciu, Tiberiu; Sur le prolongement des fonctions convexes d'ordre superieur; Bull.\ Math.\ Soc.\ Roumaine des Sc.; 36; 1934; 75--108; % B-splines with arbitrary knots. Uses `elementary function of order n' for % what we now call a `spline of degree n with simple knots'. % B-spline recurrence relations; Marsden's identity. %Popoviciu1936 % karlin/studden 23jun03 \rhl{P36} \refJ Popoviciu, Tib{\`e}re; Notes sur les fonctions convexes d'ordre sup\'erieur (I); Mathematica; 12; 1936; 81--92; %Popoviciu1937 % carl 23jun03 \rhl{P37} \refB Popoviciu, Tiberiu; Despre cea mai bun{\v a} approxima{\c t}ie a func{\c t}iilor continue prin polinoame (cinci lec{\c t}ii {\c t}inute la facultatea de {\c s}tiin{\c t}e din Cluj in anul {\c s}olar 1933--34); Institutul de Arte Grafice ``Ardealul'', Cluj (Romania); 1937; %Popoviciu1940 % carl 23jun03 \rhl{P40} \refJ Popoviciu, Tiberiu; Introduction \`a la th\'eorie des diff\'erences divis\'ees; Bull.\ Mathem., Societea Romana de Stiinte, Bukharest; 42; 1940; 65--78; % written in response to the fact that Steffensen39 failed to cite % Popoviciu33 and follow-ups, including Popoviciu's proof of the Leibniz % formula for the divided difference of a product. Also states the % `fundamental transformation formula for divided differences', something % that reduces to the formula, % \sum_{j=1}^n \psi_{1,j-1}\dvd{x_1,\ldots,x_j} + % \sum_{j>n} \psi+_{j-n+1,j-1}(x_j-x_{j-n})\dvd{x_{j-n},\ldots,x_j} % with \psi^+_{r,s}:= (\cdot-x_r)_+\cdots(\cdot-x_s)_+, for the interpolant % at the sequence x_1< x_2< \cdots that agrees, on [x_j\fromto x_{j+1}], with % the polynomial interpolant at x_{j-n+1},\ldots,x_j. Evaluation of this % formula at x_i provides a unique description of the linear functional [x_i] % in terms of the linear functionals \dvd{x_{max(1,j-n+1)},\ldots,x_{j-1}}, % j=1,2,\ldots, hence Popoviciu's name `transformation formula' for the % latter. Also proves the `mean-value formula for divided differences': % \dvd{t_0,\ldots,t_n} = \sum_j a_j(t,s)\dvd{s_j,\ldots,s_{j+n}} % for any monotone refinement s of monotone t, with a_j(t,s)\ge0 and summing % to 1. %Popoviciu1944 % carl 16aug02 \rhl{P44} \refB Popoviciu, Tib{\`e}re; Les fonctions convexes; Hermann \&\ C$^{ie}$, Paris (France); 1944; % a summary of Popoviciu's work on generalized convex functions, -- up to % that point. %Popoviciu1959a % . 14sep95 \rhl{P} \refJ Popoviciu, T.; Sur le reste dans certaines formules lin\'eaires d'approximation de l'analyse; Mathematica (Cluj); 1; 1959; 95--142; % generalises divided differences (as quotients of determinants) % to obtain a Newton form of the interpolant from a Chebyshev system. % I haven't actually looked at the paper %Popoviciu1961 % karlin/studden 25mar11 \rhl{P1961} \refJ Popoviciu, Tiberiu; Sur la conservation de l'allure de convexit\'e d'une fonction par ses polynomes d'interpolation; Mathematica; 3(26); 1961; 311--329; %Popoviciu1962 % karlin/studden 23jun03 \rhl{P62} \refJ Popoviciu, Tib{\`e}re; Sur la conservation, par le polyn\^ome d'interpolation de L. Fej\'er, du signe ou de la monotonie de la function; Analele Stiintifice; 8; 1962; 65--84; %Posdamer1982 \rhl{P} \refJ Posdamer, J. L.; Surface geometry acquisition using a binary-coded structured illlumination technique; CI???; 3; 1982; 83--92; %Posdorf1978 \rhl{P} \refD Posdorf, H.; Boolesche Methoden bei zweidimensionaler Interpolation; Siegen; 1978; %PotierVercken1994 % larry \rhl{P} \refP Potier, C., Vercken, C.; Regularity analysis of non-uniform data; \ChamonixIIb; 399--406; %PotierVerken1983 \rhl{P} \refR Potier, C., Verken, C.; Approximation de fonctions de deux variables, applications en cartographie cardiaque et en m\'et\'eorologie; Report; 1983; %Potra1980 % carl 12mar97 \rhl{P} \refJ Potra, Florian-Alexandru; A characterization of the divided differences of an operator which can be represented by Riemann integrals; Matematica -- Revue d'Analyse Num\'erique et de Th\'eorie de l'Approximation; 9(2); 1980; 251--253; % Let divided difference $T\times D\to bL(X,Z): (x,y)\mapsto[x,y]f$ % (i.e., $[x,y]f:x-y \mapsto f(x) - f(y)$) (of the map $f:T\subset X\to Z$, % $X, Z$ Bs's, $D$ convex) be Lipschitz. Then the equality % $[u,v] = 2[u,2v-u] - [v,2v-u]$ holds for all $u\not=v, 2v-u\in T$ iff % $[x,y]f = \int_0^1 Df(x+t(y-x)) dt$, all $x,y\in T$. % Also, good selection of early work on divided difference of operators. %Pottmann1991 % carl \rhl{P} \refJ Pottmann, H.; Scattered data interpolation based upon generalized minimum norm networks; \CA; 7; 1991; 247--256; %Pottmann1993a % carl \rhl{P} \refJ Pottmann, Helmut; Rational curves and surfaces with rational offsets; % TR 1, Inst.\ f\"ur Geometrie, TU Wien, 93, \CAGD; xx; 199x; xxx--xxx; %Pottmann1993b % carl \rhl{P} \refR Pottmann, Helmut; Symmetric Tchebycheffian B-spline schemes; TR 3, Inst.\ f\"ur Geometrie, TU Wien; 1993; %Pottmann1993c % carl \rhl{P} \refR Pottmann, Helmut; Developable rational B\'ezier and B-spline surfaces; TR 6, Inst.\ f\"ur Geometrie, TU Wien; 1993; %Pottmann1993d \rhl{P} \refJ Pottmann, H.; The geometry of Tchebycheffian splines; \CAGD; 10; 1993; 181--210; %Pottmann1994 % larry \rhl{P} \refP Pottmann, H.; Applications of the dual B\'ezier representation of rational curves and surfaces; \ChamonixIIa; 377--384; %Pottmann1995 % carl 19nov95 \rhl{P} \refP Pottmann, Helmut; Studying NURBS curves and surfaces with classical geometry; \Ulvik; 413--438; %PottmannDeRose1992a % . 21feb96 5dec96 \rhl{P} \refQ Pottmann, H., DeRose, T.; Classification using normal curves; (Curves and Surfaces in Computer Vision and Graphics II), Martine J. Silbermann and Hemant D. Tagare (eds.), Proc.\ SPIE {\bf 1610} (Bellingham WA); 1992; 217--228; %PottmannEck1990a % greg \rhl{P} \refJ Pottmann, H., Eck, M.; Modified multiquadric methods for scattered data interpolation over a sphere; \CAGD; 7; 1990; 313--321; %PottmannWagner1993 \rhl{P} \refR Pottmann, H., Wagner, M. G.; Helix splines as an example of affine Tchebycheffian splines; ACM; 1993; %Pouzet1980 \rhl{P} \refJ Pouzet, J.; Estimation of a surface with known discontinuities for automatic contouring purposes; Math.\ Geol.; 12; 1980; 559--575; %PowarRanaRao1989 \rhl{P} \refR Powar, P. L., Rana, S. S., Rao, R.; Local behaviour of the denominator in the construction of rational finite element basis over the rectangle in $\RR^2$; R. D. Univ., Jabalpur, India; xxx; %Powell1964a \rhl{P} \refJ Powell, M. J. D.; An efficient method of finding the minimum of a function of several variables without calculating derivatives; \CJ; 7; 1964; 155--162; %Powell1967b % larry \rhl{P} \refJ Powell, M. J. D.; On the maximum errors of polynomial approximations defined by interpolation and least squares criteria; \CJ; 9; 1967; 404--407; %Powell1968 % sherm, proceedings update (was added to proceed.tex) \rhl{P} \refP Powell, M. J. D.; On best $L_2$ spline approximations; \Collatz; 317--339; %Powell1969a \rhl{P} \refQ Powell, M. J. D.; A comparison of spline approximations with classical interpolation methods; (Proc.\ IFIP 68), xxx (ed.), North Holland (Amsterdam); 1969; 95--98; %Powell1969b % larry \rhl{P} \refJ Powell, M. J. D.; The local dependence of least squares cubic splines; \SJNA; 6; 1969; 398--413; % also considers smoothing splines a la %Powell70 %Powell1970 % larry \rhl{P} \refP Powell, M. J. D.; Curve fitting by splines in one variable; \Hayes; 65--83; % uses cubic splines, a weighted distance measure, and measures roughness % (to be minimized) by a weighted sum of squares of jumps in third derivative % (which for the particular weights ultimately chosen, looks a bit like an % approximation to the integral of the square of the fourth derivative). %Powell1972 % author 20jun97 \rhl{P} \refR Powell, M. J. D.; A Fortran subroutine for calculating a cubic spline approximation to a given function; AERE Report No.~R.7308 (Harwell Laboratory); 1972; %Powell1974 \rhl{P} \refP Powell, M. J. D.; Piecewise quadratic surface fitting for contour plotting; \Evans; 253--271; %Powell1977 % sherm, editor update \rhl{P} \refQ Powell, M. J. D.; Numerical methods for fitting functions of two variables; (The State of the Art in Numerical Analysis), D. Jacobs (ed.), Academic Press (New York); 1977; 563--604; %Powell1981 % carl 20jun97 \rhl{P} \refB Powell, M. J. D.; Approximation Theory and Methods; Cambridge University Press (Cambridge, England); 1981; % ix+339pp %Powell1987 \rhl{P} \refP Powell, M. J. D.; Radial basis functions for multivariable interpolation: a review; \ShrivenhamI; 143--167; %Powell1987b \rhl{P} \refP Powell, M. J. D.; Radial basis function approximations to polynomials; \Griffithseise; 223--241; %Powell1989 \rhl{P} \refR Powell, M. J. D.; TOLMIN: A Fortran package for linearly constrained optimization calculations; Univ.\ Cambridge; 1989; %Powell1990a \rhl{P} \refR Powell, M. J. D.; The theory of radial basis function approximation in 1990; DAMTP rep.\ NA11, University of Cambridge; 1990; % published; see 92b %Powell1990b \rhl{P} \refP Powell, M. J. D.; Univariate multiquadric approximation: reproduction of linear polynomials; \Duisburg; 227--240; %Powell1991 % larry \rhl{P} \refP Powell, M. J. D.; Univariate multiquadric interpolation: some recent results; \ChamonixI; 371--382; %Powell1992a % carl 20jun97 \rhl{P} \refP Powell, M. J. D.; Tabulation of thin plate splines on a very fine two-dimensional grid; \Nmatnion; 221--244; % DAMTP 1992/NA2, Feb.: 1992: %Powell1992b % author \rhl{P} \refQ Powell, M. J. D.; The theory of radial basis function approximation in 1990; (Advances in Numerical Analysis II: wavelets, subdivision algorithms and radial functions), W. A. Light (ed.), Clarendon Press (Oxford); 1992; 105--210; %Powell1993a % carl \rhl{P} \refJ Powell, M. J. D.; Truncated Laurent expansions for the fast evaluation of thin plate splines; \NA; 5; 1993; 99--120; %Powell1994a % yegorov 14sep95 \rhl{P} \refJ Powell, M. J. D.; The uniform convergence of thin plate spline interpolation in two dimensions; \NM; 68; 1994; 107--128; %Powell1994b % author 20jun97 \rhl{P} \refR Powell, M. J. D.; A `taut string algorithm' for straightening a piecewise linear path in two dimensions; Report No.~DAMTP 1994/NA7 (University of Cambridge); 1994; % to appear in \IMAJNA %Powell1994c % author 20jun97 \rhl{P} \refQ Powell, M. J. D.; Some algorithms for thin plate spline interpolation to functions of two variables; (Advances in Computational Mathematics: New Delhi, India), H. P. Dikshit and C. A. Micchelli (eds.), World Scientific (Singapore); 1994; 303--319; %Powell1995 % peters 12mar97 \rhl{P} \refP Powell, M. J. D.; An algorithm that straightens and smooth piecewise linear curves in two dimensions; \Ulvik; 439--454; %Powell1996 % . 12mar97 \rhl{P} \refR Powell, M. J. D.; A review of methods for multivariable interpolation at scattered data points; DAMTP NA/11, Cambridge (England); 1996; % invited talk given at Dundee %Powell1996b % author 20jun97 \rhl{P} \refQ Powell, M. J. D.; A thin plate spline method for mapping curves into curves in two dimensions; (Computational Techniques and Applications: CTAC95), R. L. May and A. K. Easton (eds.), World Scientific (Singapore); 1996; 43--57; %Powell1997 % author 20jun97 \rhl{P} \refJ Powell, M. J. D.; A new iterative algorithm for thin plate spline interpolation in two dimensions; \AoNM; 4; 1997; 519--527; %Powell1998 % carl 24mar99 \rhl{P} \refR Powell, M. J. D.; Recent research in Cambridge on radial basis functions; DAMPTP report 1998/NA5, Cambridge (England); 1998; %PowellSabin1977 % larry \rhl{P} \refJ Powell, M. J. D., Sabin, M. A.; Piecewise quadratic approximations on triangles; \ACMTMS; 3; 1977; 316--325; %PrasadVarma1979 % carl \rhl{P} \refJ Prasad, J., Varma, A. K.; Lacunary interpolation by quintic splines; \SJNA; 16; 1979; 1075--1079; %Pratt1985 % greg \rhl{P} \refJ Pratt, M. J.; Smooth parametric surface approximations to discrete data; \CAGD; 2; 1985; 165--171; %Pratt1986 % carl \rhl{P} \refP Pratt, M. J.; Parametric curves and surfaces as used in computer aided design; \SurfacesI; 19--45; %Pratt1990a % greg \rhl{P} \refJ Pratt, M. J.; Cyclides in computer aided geometric design; \CAGD; 7; 1990; 221--242; %Pratt1993 % carl \rhl{P} \refP Pratt, Michael J.; Geometric methods for Computer-Aided Design; \Piegl; 271--320; % expository %PrattGeisow1986 % carl \rhl{P} \refP Pratt, M. J., Geisow, A. D.; Surface/surface intersection problems; \SurfacesI; 117--142; %Prautzsch1983 % larry \rhl{P} \refR Prautzsch, H.; Unterteilungsalgorithmen f\"ur B\'ezier und B-spline Fl\"achen; Diplom Thesis, Univ.\ Braunschweig; 1983; %Prautzsch1984a % greg \rhl{P} \refJ Prautzsch, H.; A short proof of the Oslo algorithm; \CAGD; 1; 1984; 95--96; %Prautzsch1984b % greg \rhl{P} \refJ Prautzsch, H.; Degree elevation of B-spline curves; \CAGD; 1; 1984; 193--198; %Prautzsch1984c \rhl{P} \refD Prautzsch, H.; Unterteilungsalgorithmen f\"ur multivariate Splines -- ein geometrischer Zugang; Univ.\ Braunschweig; 1984; %Prautzsch1985 % greg \rhl{P} \refJ Prautzsch, H.; Generalized subdivision and convergence; \CAGD; 2; 1985; 69--75; %Prautzsch1986 \rhl{P} \refJ Prautzsch, H.; The location of the control points in the case of box splines; \IMAJNA; 6; 1986; 43--49; %Prautzsch1988b \rhl{P} \refR Prautzsch, H.; The generation of box spline surfaces; CA; 1988; %Prautzsch1991a % sherm, checked journal for this, earlier ref, Prautzsch88a had CAGD:5:1988 \rhl{P} \refJ Prautzsch, H.; Linear subdivision; \LAA; 143; 1991; 223--230; %Prautzsch1992 % LLS Lai-Schumaker book \rhl{Pra92} \refJ Prautzsch, H.; On convex B\'ezier triangles; \RAIROAN; 26; 1992; 23--36; %Prautzsch1993a % carl \rhl{P} \refJ Prautzsch, Hartmut; Algorithmic blending; \JAT; 72(1); 1993; 87--102; %PrautzschBoehmPaluszny2002 % carl 23jun03 \rhl{} \refB Prautzsch, H., Boehm, W., Paluszny, M.; B\'ezier and B-spline Techniques; Springer (Berlin); 2002; % ISBN 3-450-43761-4 % visualization %PrautzschKobbelt1994 % author 22may98 \rhl{P} \refJ Prautzsch, H., Kobbelt, L.; Convergence of subdivision and degree elevation; \AiCM; 2; 1994; 143--154; %PrautzschMicchelli1987 % greg \rhl{P} \refJ Prautzsch, H., Micchelli, C. A.; Computing curves invariant under halving; \CAGD; 4; 1987; 133--140; %PrautzschUmlauf2000 % larry 20apr00 \rhl{} \refP Prautzsch, Hartmut, Umlauf, Georg; Triangular $g^2$-splines; \Stmalod; 335--342; %Prenter1971 \rhl{P} \refR Prenter, P. M.; A method of collocation for the numerical solution of integral equations of the section kind; xx; 19xx; %Prenter1971b % carl \rhl{P} \refJ Prenter, P. M.; Piecewise L-splines; \NM; 18; 1971; 243--253; % AyalonDynLevin09 points out that the error formula for splines in tension % is incorrect (ignores boundary contributions from integration by parts. %Prenter1971c % . 29apr97, 28sep12 \rhl{P} \refJ Prenter, P. M.; Lagrange and Hermite interpolation in Banach spaces; \JAT; 4(4); 1971; 419--432; % defines a `polynomial' on the Bs X to the Bs Y as the map $x\mapsto \sum_j % L_jx^j, with L_jx^j := L_j(x,\ldots,x) and L_j a linear map from X^j to Y. % Strange. %Prenter1975 \rhl{P} \refB Prenter, P. M.; Splines and Variational Methods; Wiley (New York); 1975; %PreparataShamos1985 % larry Lai-Schumaker book \rhl{PreS85} \refB Preparata, F. P., Shamos, M. I.; Computational Geometry: An Introduction; Spring\-er-Verlag (New York); 1985; %PressFlanneryTeukolskyVetterling1988a \rhl{P} \refB Press, W., Flannery, B., Teukolsky, S., Vetterling, W.; Numerical Recipes in C; Cambridge University Press (Cambridge); 1988; %PrestinQuak1991 \rhl{P} \refR Prestin, J., Quak, E.; On interplation and best one-sided approximation by splines in $L^p$; CAT Report 246, Memphis; 1991; %PrestinQuak1993 \rhl{P} \refR Prestin, J., Quak, E.; Trigonometric interpolation and wavelet decompositions; CAT 296; 1993; %PrestinQuak1994 % larry \rhl{P} \refP Prestin, J., Quak, E.; A duality principle for trigonometric wavelets; \ChamonixIIb; 407--418; %PretoriusEyre1987 \rhl{P} \refJ Pretorius, L., Eyre, D.; Spline-Gauss rules and the Nystrom method for solving integral equations in quantum scattering; \JCAM; 18; 1987; 235--248; %Preusser1984 \rhl{P} \refJ Preusser, A.; Algorithm 626: TRICP: A contour plot program for triangular meshes; \ACMTMS; 11; 1984; 473--475; %Preusser1984b \rhl{P} \refJ Preusser, A.; Computing contours by successive solution of quintic polynomial equations; \ACMTMS; 11; 1984; 463--472; %Preusser1985 \rhl{P} \refJ Preusser, A.; Remark on Algorithm 526; \ACMTMS; 12; 1985; 186--187; %PriceCavendishVarga1968 \rhl{P} \refR Price, H. S., Cavendish, J. C., Varga, R. S.; Numerical methods of higher order accuracy for diffusion convection equations; TAMS 243; 19xx; %PriceSimonsen1962 \rhl{P} \refR Price, J. F., Simonsen, R. H.; Various methods and computer routines for approximation, curve fitting, and interpolation; Boeing Labs.; 1962; %PriceVarga1970 % larry \rhl{P} \refQ Price, H. S., Varga, R. S.; Error bounds for semidiscrete Galerkin approximations of parabolic problems with applications to petroleum reservoir mechanics; (Numerical Solution of Field Problems in Continuum Physics), xxx (ed.), AMS (Providence RI); 1970; 74--94; %Priver1970 \rhl{P} \refD Priver, A. S.; Data smoothing in interactive computer graphics; Harvard Univ.; 1970; %Prolla1986 \rhl{P} \refP Prolla, J. B.; Approximation by positive elements of subalgebras of real valued functions; \Chile; 185--192; %Proriol1957 % shayne 16mar01 \rhl{} \refJ Proriol, J.; Sur une famille de polynomes \`a deux variables orthogonaux dans un triangle; \CRASP; 245; 1957; 2459--2461; %Prossdorf1984a % . 05feb96 \rhl{P} \refJ Pr\"o{\ss}dorf, S.; Ein Lokalisierungsprinzip in der Theorie der Spline-Approximationen und einige Anwendungen; \MNa; 119; 1984; 239--255; %ProssdorfRathsfeld1984 \rhl{P} \refR Pr\"ossdorf, S., Rathsfeld, A.; On spline Galerkin methods for singular integral equations with piecewise continouus coefficients; E. Berlin; 1984; %ProssdorfRathsfeld1989 % . 12mar97 \rhl{P} \refJ Pr{\"o}ssdorf, S., Rathsfeld, A.; Quadrature and collocation methods for singular integral equations on curves with corners; Z. Anal.\ Anwendungen; 8; 1989; 197--220; %ProssdorfSchneider % . 05feb96 \rhl{P} \refJ Pr\"o{\ss}dorf, S., Schneider, H.; Spline approximation methods for multidimensional periodic pseudodifferential equations; Integral Equations Operator Theory; 15; 1992; 626--672; %ProssdorfSloan1992 % carl \rhl{P} \refJ Pr\"ossdorf, S., Sloan, I.; Quadrature method for singular integral equations on closed curves; \NM; 61; 1992; 543--561; %Pruess1976 % sonya \rhl{P} \refJ Pruess, S.; Properties of splines in tension; \JAT; 17; 1976; 86--96; % see Grandison97 %Pruess1978 \rhl{P} \refJ Pruess, S.; An algorithm for computing smoothing splines in tension; \C; 19; 1978; 365--373; %Pruess1979 % larry, carl \rhl{P} \refJ Pruess, Steven; Alternatives to the exponential spline in tension; \MC; 33(148); 1979; 1273--1281; %Pruess1986a % author \rhl{P} \refJ Pruess, S.; Interpolation schemes for collocation solutions of two point boundary value problems; \SJSSC; 7; 1986; 322--333; % local interpolation %Pruess1988a % sonya \rhl{P} \refJ Pruess, S.; Stability bounds for local Lagrangian interpolation; \JAT; 53; 1988; 117--127; %Pruess1993 % . \rhl{P} \refJ Pruess, S.; Shape preserving $C^2$ cubic spline interpolation; \IMAJNA; 13 (4); 1993; 493--507; %Pruess1999 \rhl{P} \refR Pruess, S.; Estimating the eigenvalues of Sturm-Liouville problems by approximating the differential equation; Univ.\ New Mexico; 1999; %PruessXX \rhl{P} \refJ Pruess, S.; Local bases for computing smoothing splines in tension; \CJ; X; 19XX; XX; %Pualtuanea1983 % shayne 21jan02 \rhl{} \refJ P{\u a}lt{\u a}nea, R.; Sur un op\'erateur polynomial d\'efini sur l'ensemble des fonctions int\'egrables; Univ.{} "Babe\c s-Bolyai", Cluj-Napoca; 83--2; 1983; 101--106; % Defines the Bernstein-Durrmeyer operator with Jacobi weights %Pudlatz1970 % larry \rhl{P} \refD Pudlatz, H.; Normalit\"at und Stetigkeit bei der Tschebyscheff-Approximation mit zeichenregul\"aren $\gamma$-Polynomen; Univ.\ M\"unster; 1970; %PuechChasseryPitas1997 % larry 10sep99 \rhl{PCP} \rhl{P} \refP Puech, W., Chassery, J.-M., Pitas, I.; Curved surfaces reconstruction based on parallels; \ChamonixIIIa; 363--370; %PychTaberska1997 % shayne 26aug98 \rhl{P} \refJ Pych--Taberska, P.; Rate of pointwise convergence of Bernstein polynomials for some absolutely continuous functions; \JMAA; 212; 1997; 9--19;