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12
H.: A set of symmetric quadrature rules on triangles and tetrahedra
 J. Comput. Math
, 2009
"... We present a program for computing symmetric quadrature rules on triangles and tetrahedra. A set of rules are obtained by using this program. Quadrature rules up to order 21 on triangles and up to order 14 on tetrahedra have been obtained which are useful for use in finite element computations. All ..."
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We present a program for computing symmetric quadrature rules on triangles and tetrahedra. A set of rules are obtained by using this program. Quadrature rules up to order 21 on triangles and up to order 14 on tetrahedra have been obtained which are useful for use in finite element computations. All rules presented here have positive weights with points lying within the integration domain.
DISCRETE FOURIER ANALYSIS ON FUNDAMENTAL DOMAIN OF Ad LATTICE AND ON SIMPLEX IN dVARIABLES
, 809
"... Abstract. A discrete Fourier analysis on the fundamental domain Ωd of the ddimensional lattice of type Ad is studied, where Ω2 is the regular hexagon and Ω3 is the rhombic dodecahedron, and analogous results on ddimensional simplex are derived by considering invariant and antiinvariant elements. ..."
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Abstract. A discrete Fourier analysis on the fundamental domain Ωd of the ddimensional lattice of type Ad is studied, where Ω2 is the regular hexagon and Ω3 is the rhombic dodecahedron, and analogous results on ddimensional simplex are derived by considering invariant and antiinvariant elements. Our main results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange interpolation on the simplex is shown to satisfy an explicit compact formula and the Lebesgue constant of the interpolation is shown to be in the order of (log n) d. The basic trigonometric functions on the simplex can be identified with Chebyshev polynomials in several variables already appeared in literature. We study common zeros of these polynomials and show that they are nodes for a family of Gaussian cubature formulas, which provides only the second known example of such formulas.
DISCRETE FOURIER ANALYSIS ON A DODECAHEDRON AND A TETRAHEDRON
"... Abstract. A discrete Fourier analysis on the dodecahedron is studied, from which results on a tetrahedron are deduced by invariance. The results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange interpol ..."
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Abstract. A discrete Fourier analysis on the dodecahedron is studied, from which results on a tetrahedron are deduced by invariance. The results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange interpolation on the tetrahedron is shown to satisfy an explicit compact formula and the Lebesgue constant of the interpolation is shown to be in the order of (log n) 3. 1.
Cubature formula and interpolation on the cubic domain
 NUMER. MATH.: THEORY METHOD APPL., ACCEPTED FOR PUBLICATION
, 2008
"... Several cubature formulas on the cubic domains are derived using the discrete Fourier analysis associated with lattice tiling, as developed in [10]. The main results consist of a new derivation of the Gaussian type cubature for the product Chebyshev weight functions and associated interpolation pol ..."
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Cited by 3 (3 self)
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Several cubature formulas on the cubic domains are derived using the discrete Fourier analysis associated with lattice tiling, as developed in [10]. The main results consist of a new derivation of the Gaussian type cubature for the product Chebyshev weight functions and associated interpolation polynomials on [−1,1] 2, as well as new results on [−1,1]³. In particular, compact formulas for the fundamental interpolation polynomials are derived, based on n 3 /4 + O(n²) nodes of a cubature formula on [−1,1]³.
FOURIER SERIES AND APPROXIMATION ON HEXAGONAL AND TRIANGULAR DOMAINS
, 802
"... Abstract. Several problems on Fourier series and trigonometric approximation on a hexagon and a triangle are studied. The results include Abel and Cesàro summability of Fourier series, degree of approximation and best approximation by trigonometric functions, both direct and inverse theorems. One of ..."
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Abstract. Several problems on Fourier series and trigonometric approximation on a hexagon and a triangle are studied. The results include Abel and Cesàro summability of Fourier series, degree of approximation and best approximation by trigonometric functions, both direct and inverse theorems. One of the objective of this study is to demonstrate that Fourier series on spectral sets enjoy a rich structure that allow an extensive theory for Fourier expansions and approximation. 1.
COMPLEX VERSUS REAL ORTHOGONAL POLYNOMIALS OF TWO VARIABLES
"... Abstract. Orthogonal polynomials of two real variables can often be represented in complex variables. We explore the connection between the two types of representations and study the structural relations of complex orthogonal polynomials. The complex Hermite orthogonal polynomials and the disk pol ..."
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Abstract. Orthogonal polynomials of two real variables can often be represented in complex variables. We explore the connection between the two types of representations and study the structural relations of complex orthogonal polynomials. The complex Hermite orthogonal polynomials and the disk polynomials are used as illustrating examples. 1.
Discrete Fourier analysis with lattices on planar domains
 Numer. Algorithms
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Discrete Fourier Analysis and Chebyshev Polynomials with G2 Group
"... Abstract. The discrete Fourier analysis on the 300 –600–900 triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group G2, which leads to the definition of four families generalized Chebyshev polynomials. The study of these polynomial ..."
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Abstract. The discrete Fourier analysis on the 300 –600–900 triangle is deduced from the corresponding results on the regular hexagon by considering functions invariant under the group G2, which leads to the definition of four families generalized Chebyshev polynomials. The study of these polynomials leads to a Sturm–Liouville eigenvalue problem that contains two parameters, whose solutions are analogues of the Jacobi polynomials. Under a concept of mdegree and by introducing a new ordering among monomials, these polynomials are shown to share properties of the ordinary orthogonal polynomials. In particular, their common zeros generate cubature rules of Gauss type. Key words: discrete Fourier series; trigonometric; group G2; PDE; orthogonal polynomials 2010 Mathematics Subject Classification: 41A05; 41A10 1
DISCRETE FOURIER ANALYSIS ON A DODECAHEDRON AND
, 803
"... Abstract. A discrete Fourier analysis on the dodecahedron is studied, from which results on a tetrahedron is deduced by invariance. The results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange interpola ..."
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Abstract. A discrete Fourier analysis on the dodecahedron is studied, from which results on a tetrahedron is deduced by invariance. The results include Fourier analysis in trigonometric functions, interpolation and cubature formulas on these domains. In particular, a trigonometric Lagrange interpolation on the tetrahedron is shown to satisfy an explicit compact formula and the Lebesgue constant of the interpolation is shown to be in the order of (log n) 3. 1.
GENERALIZED CHARACTERISTIC POLYNOMIALS AND GAUSSIAN CUBATURE RULES
"... Abstract. Generalized characteristic polynomial of a family of near banded Topelitz matrices are shown to be orthogonal polynomials of two variables and they possess maximal number of real common zeros, which generate a family of Gaussian cubature rules in two variables. 1. ..."
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Abstract. Generalized characteristic polynomial of a family of near banded Topelitz matrices are shown to be orthogonal polynomials of two variables and they possess maximal number of real common zeros, which generate a family of Gaussian cubature rules in two variables. 1.