Orthogonal polynomials and cubature formulae on spheres and on simplices

Xu, Yuan

doi:10.4310/maa.1998.v5.n2.a5

Cited by 29 publications

(27 citation statements)

References 21 publications

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“…Such a basis is indeed homogeneous; we showed in [8] that it can be obtained from a Z of R n k is n. Then it is easy to verify, using (2.4), that the polynomials…”

Section: And Only If P Lies In the Intervalmentioning

confidence: 98%

A note on summability of multiple Laguerre expansions

2000

Proc. Amer. Math. Soc.

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Abstract. A simple structure of the multiple Laguerre polynomial expansions is used to study the Cesàro summability above the critical index for the convolution type Laguerre expansions. The multiple Laguerre polynomial expansion of an 1 -radial function f 0 (|x|) is shown to be an 1 -radial function that coincides with the Laguerre polynomial expansion of f 0 , which allows us to settle the problem of summability below the critical index for the 1 -radial functions.

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“…Such a basis is indeed homogeneous; we showed in [8] that it can be obtained from a Z of R n k is n. Then it is easy to verify, using (2.4), that the polynomials…”

Section: And Only If P Lies In the Intervalmentioning

confidence: 98%

A note on summability of multiple Laguerre expansions

2000

Proc. Amer. Math. Soc.

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“…In [22] it is shown that any cubature formula for W 0 with all nodes inside Σ 2 corresponds to a cubature formula on the sphere S 2 . In this case, the reproducing kernel takes a simple form…”

Section: Reproducing Kernelmentioning

confidence: 99%

Constructing cubature formulae by the method of reproducing kernel

2000

Numerische Mathematik

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“…This formula can be easily verified by changing variables (cf. [3], [12]). It implies, in particular, that…”

Section: Orthogonal Structure and Compact Formulae For Reproducing Kementioning

confidence: 99%

Summability of Fourier orthogonal series for Jacobi weight functions on the simplex in ℝ^{𝕕}

1998

Proc. Amer. Math. Soc.

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Abstract. We study the Fourier expansion of a function in orthogonal polynomial series with respect to the weight functionsIt is proved that such an expansion is uniformly (C, δ) summable on the simplex for any continuous function if and only if δ > |α| 1 + (d − 1)/2. Moreover, it is shown that (C, |α| 1 + (d + 1)/2) means define a positive linear polynomial identity, and the index is sharp in the sense that (C, δ) means are not positive for 0 < δ < |α| 1 + (d + 1)/2.

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Orthogonal polynomials and cubature formulae on spheres and on simplices

Cited by 29 publications

References 21 publications

A note on summability of multiple Laguerre expansions

A note on summability of multiple Laguerre expansions

Constructing cubature formulae by the method of reproducing kernel

Summability of Fourier orthogonal series for Jacobi weight functions on the simplex in ℝ^{𝕕}

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