On minimal cubature formulae for product weight functions

Bojanov, Borislav; Petrova, Guergana

doi:10.1016/s0377-0427(97)00133-7

Cited by 22 publications

(43 citation statements)

References 4 publications

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“…The first family is given by the "MorrowPatterson-Xu" points, which have been studied in the contexts of minimal cubature [27,17,44,1], interpolation [44,2] and hyperinterpolation [11,9,13]. The second family, termed the "Padua points", has been recently studied in the interpolation context [10,3,5,12,14].…”

Section: Nontensorial Clenshaw-curtis Cubaturementioning

confidence: 99%

“…(20)) for odd n, and almost minimal (up to 1 node) for even n. We have termed them after Morrow and Patterson [27], who gave originally the explicit formula in the odd case, and Xu [44], who obtained the explicit formula for even instances ("even" and "odd" are interchanged here with respect the usual setting, which refers to degree of exactness 2n − 1). Formulas of this type, even in a more general setting, have been studied by various other authors, for example in [17,1]. The MPX (Morrow-Patterson-Xu) points are defined as union of the bidimensional Chebyshev-like grids…”

Section: The Morrow-patterson-xu Pointsmentioning

confidence: 99%

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Nontensorial Clenshaw–Curtis cubature

2008

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We extend Clenshaw-Curtis quadrature to the square in a nontensorial way, by using Sloan's hyperinterpolation theory and two families of points recently studied in the framework of bivariate (hyper)interpolation, namely the Morrow-Patterson-Xu points and the Padua points. The construction is an application of a general approach to product-type cubature, where we prove also a relevant stability theorem. The resulting cubature formulas turn out to be competitive on nonentire integrands with tensorproduct Clenshaw-Curtis and Gauss-Legendre formulas, and even with the few known minimal formulas.2000 AMS subject classification: 65D32.

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Section: Nontensorial Clenshaw-curtis Cubaturementioning

confidence: 99%

Section: The Morrow-patterson-xu Pointsmentioning

confidence: 99%

Nontensorial Clenshaw–Curtis cubature

2008

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“…It suffices to establish (15) for f ∈ {T k : |k| ≤ 2n − 1}, since this set is an orthogonal basis of Π d n . In this case we have For the case of d = 2, Theorem 2.2 contains two distinct cubature formulae for σ = (E, E), (E, O), respectively, whose number of nodes are either equal to N * in (10) or N * + 1, those are the ones that have appeared in [14,22], and later in [1], as mentioned earlier. For d = 3, there are 4 distinct formulae for…”

Section: Introductionmentioning

confidence: 96%

“…A close inspection of the factorization method shows that it actually allows us to derive cubature formulae of degree 2n − 1 for W d with roughly 2(n/2) d many nodes. This number of nodes is substantially less than n d of the product cubature formula or n d /2 d/2 of the formulae in [1], although it likely far from optimal as seen from (3).…”

Section: Introductionmentioning

confidence: 96%

New cubature formulae and hyperinterpolation in three variables

Marchi

Vianello

2009

Bit Numer Math

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A new algebraic cubature formula of degree 2n + 1 for the product Chebyshev measure in the d-cube with ≈ n d /2 d−1 nodes is established. The new formula is then applied to polynomial hyperinterpolation of degree n in three variables, in which coefficients of the product Chebyshev orthonormal basis are computed by a fast algorithm based on the 3-dimensional FFT. Moreover, integration of the hyperinterpolant provides a new Clenshaw-Curtis type cubature formula in the 3-cube.

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“…Thus the number of nodes in (3.8) is minimal when m is even and k = 1 and is at most one more than the minimal number of nodes otherwise. Theorem 3.3 and a similar theorem of Bojanov and Petrova [3] combine to give a simpler formula for the coefficients c n,q in (3.3).…”

Section: Interpolation At the Geronimus Nodesmentioning

confidence: 90%

Bivariate Polynomial Interpolation at the Geronimus Nodes

Harris¹

2013

Complex Analysis and Dynamical Systems V

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Abstract. We consider a class of orthogonal polynomials that satisfy a threeterm recurrence formula with constant coefficients. This class contains the Geronimus class and, in particular, all four kinds of the Chebyshev polynomials. There are alternation points for each of these orthogonal polynomials that have a special compatibility with the polynomials of lower index. These points are the coordinates of two sets of nodes in R 2 , which we call Geronimus nodes. The Chebyshev nodes considered separately by Yuan Xu and the author are a special case.We obtain an explicit formula (involving the reproducing kernel) for bivariate Lagrange polynomials for the Geronimus nodes and we apply this to obtain a bivariate interpolation theorem and a cubature formula. These theorems are a consequence of a surprisingly elementary connection between Lagrange polynomials, interpolation formulas and cubature formulas, which we explain in an appendix. Finally, we discuss a general bivariate Markov theorem where polynomials in the Geronimus class are extremal.

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On minimal cubature formulae for product weight functions

Cited by 22 publications

References 4 publications

Nontensorial Clenshaw–Curtis cubature

Nontensorial Clenshaw–Curtis cubature

New cubature formulae and hyperinterpolation in three variables

Bivariate Polynomial Interpolation at the Geronimus Nodes

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