Cubature formulae and orthogonal polynomials

[1]  P. Appell,et al.  Sur une classe de polynômes à deux variables et le calcul approché des intégrales doubles , 2022 .

[2]  P. Appell,et al.  Fonctions hypergéométriques et hypersphériques : polynomes d'Hermite , 1926 .

[3]  D. Jackson.  Formal properties of orthogonal polynomials in two variables , 1936 .

[4]  J. Radon,et al.  Zur mechanischen Kubatur , 1948 .

[5]  W. Gröbner,et al.  Über die Konstruktion von Systemen orthogonaler Polynome in ein- und zwei-dimensionalen Bereichen , 1948 .

[6]  Morris Weisfeld,et al.  Orthogonal polynomials in several variables , 1959, Numerische Mathematik.

[7]  I. P. Mysovskikh.  Proof of the minimality of the number of nodes in the cubature formula for a hypersphere , 1966 .

[8]  I. P. Mysovskii.  Radon's paper on the cubature formula , 1967 .

[9]  H. L. Krall,et al.  Orthogonal polynomials in two variables , 1967 .

[10]  A. Stroud.  Integration Formulas and Orthogonal Polynomials. II , 1967 .

[11]  Philip Rabinowitz,et al.  Methods of Numerical Integration , 1985 .

[12]  I. P. Mysovskikh.  The construction of cubature formulae and orthogonal polynomials , 1967 .

[13]  P. M. Hirsch.  Evaluation of orthogonal polynomials and relationship to evaluating multiple integrals , 1968 .

[14]  A. Stroud.  Integration Formulas and Orthogonal Polynomials for Two Variables , 1969 .

[15]  I. P. Mysovskikh.  Cubature formulae and orthogonal polynomials , 1970 .

[16]  F. Fritsch.  On the existence of regions with minimal third degree integration formulas , 1970 .

[17]  R. Franke.  Obtaining cubatures for rectangles and other planar regions by using orthogonal polynomials , 1971 .

[18]  I. P. Mysovskikh.  The application of orthogonal polynomials to cubature formulae , 1972 .

[19]  R. Franke,et al.  Minimal Point Cubatures of Precision Seven for Symmetric Planar Regions , 1973 .

[20]  A. Stroud.  Approximate calculation of multiple integrals , 1973 .

[21]  Quadrature Formulas over Fully Symmetric Planar Regions , 1973 .

[22]  Local reality on algebraic varieties , 1974 .

[23]  C. Günther.  Third Degree Integration Formulas with Four Real Points and Positive Weights in Two Dimensions , 1974 .

[24]  Cubature formulas of degree eleven for symmetric planar regions , 1975 .

[25]  G. N. Gegel.  A quadrature formula for a four-dimensional sphere☆ , 1975 .

[26]  Anny Haegemans,et al.  Cubature formulas of degree nine for symmetric planar regions , 1975 .

[27]  Hans Michael Möller,et al.  Hermite interpolation in several variables using ideal-theoretic methods , 1976, Constructive Theory of Functions of Several Variables.

[28]  Circularly symmetrical integration formulas for two-dimensional circularly symmetrical regions , 1976 .

[29]  H. Michael Möller,et al.  Mehrdimensionale Hermite-Interpolation und numerische Integration , 1976 .

[30]  Robert Piessens,et al.  Construction of cubature formulas of degree eleven for symmetric planar regions, using orthogonal polynomials , 1976 .

[31]  H. M. Möller,et al.  Kubaturformeln mit minimaler Knotenzahl , 1976 .

[32]  A. Haegemans.  Tables of symmetrical cubature formulas for the two-dimensional hexagon , 1976 .

[33]  Anny Haegemans,et al.  Construction of Cubature Formulas of Degree Seven and Nine Symmetric Planar Regions, Using Orthogonal Polynomials , 1977 .

[34]  T. N. L. Patterson,et al.  Construction of Algebraic Cubature Rules Using Polynomial Ideal Theory , 1978 .

[35]  H. Schmid.  On cubature formulae with a minimal number of knots , 1978 .

[36]  H. M. Möller.  The Construction of Cubature Formulae and Ideals of Principal Classes , 1979 .

[37]  H. M. Möller,et al.  Lower Bounds for the Number of Nodes in Cubature Formulae , 1979 .

[38]  Interpolatorische Kubaturformeln und reelle Ideale , 1980 .

[39]  H. Engels,et al.  Numerical Quadrature and Cubature , 1980 .

[40]  Franz Peherstorfer,et al.  Characterization of Positive Quadrature Formulas , 1981 .

[41]  G. Wanner,et al.  The number of positive weights of a quadrature formula , 1982 .

[42]  M. Kowalski,et al.  The Recursion Formulas for Orthogonal Polynomials in n Variables , 1982 .

[43]  A. Haegemans.  Construction of Known and New Cubature Formulas of Degree Five for Three-Dimensional Symmetric Regions, Using Orthogonal Polynomials , 1982 .

[44]  M. Kowalski,et al.  Orthogonality and Recursion Formulas for Polynomials in n Variables , 1982 .

[45]  Franz Peherstorfer.  Characterization of Quadrature Formula II , 1984 .

[46]  Algebraic characterization of orthogonality in the space of polynomials , 1985 .

[47]  Ronald Cools,et al.  Construction of Symmetric Cubature Formulae with the Number of Knots (Almost) Equal to Möller’s Lower Bound , 1987 .

[48]  H. Michael Möller,et al.  On the Construction of Cubature Formulae with Few Nodes Using Groebner Bases , 1987 .

[49]  H. Schmid.  On Minimal Cubature Formulae of Even Degree , 1988 .

[50]  Why do so many cubature formulae have so many positive weights? , 1988 .

[51]  Ronald Cools,et al.  On cubature formulae of degree 4k+1 attaining Möller's lower bound for integrals with circular symmetry , 1992 .

[52]  H. Berens,et al.  On the Number of Nodes of Odd Degree Cubature Formulae for Integrals with Jacobi Weights on a Simplex , 1992 .

[53]  Terje O. Espelid,et al.  Numerical Integration: Recent Developments, Software and Applications. , 1993 .

[54]  R. Cools,et al.  Monomial cubature rules since “Stroud”: a compilation , 1993 .

[55]  Yuan Xu,et al.  On multivariate orthogonal polynomials , 1993 .

[56]  Ronald Cools,et al.  A new lower bound for the number of nodes in cubature formulae of degree 4 n + 1 for some circularly symmetric integrals , 1993 .

[57]  Yuan Xu,et al.  A characterization of positive quadrature formulae , 1994 .

[58]  Yuan Xu,et al.  On bivariate Gaussian cubature formulae , 1994 .

[59]  Yuan Xu,et al.  Block Jacobi matrices and zeros of multivariate orthogonal polynomials , 1994 .

[60]  Yuan Xu.  Recurrence formulas for multivariate orthogonal polynomials , 1994 .

[61]  Yuan Xu,et al.  Multivariate Gaussian cubature formulae , 1995 .

[62]  Hans Joachim Schmid,et al.  Two-Dimensional Minimal Cubature Formulas and Matrix Equations , 1995, SIAM J. Matrix Anal. Appl..

[63]  Yuan Xu,et al.  On two-dimensional definite orthogonal systems and a lower bound for the number of nodes of associated cubature formulae , 1995 .

[64]  Ronald Cools,et al.  Constructing cubature formulae: the science behind the art , 1997, Acta Numerica.

[65]  S. L. Sobolev,et al.  Theory of Cubature Formulas , 1997 .

[66]  S. L. Sobolev,et al.  Problems and Results of the Theory of Cubature Formulas , 1997 .

[67]  R. Cools.  Monomial cubature rules since “Stroud”: a compilation—part 2 , 1999 .