The Lebesgue constants for cardinal spline interpolation

Richards, F. B.

doi:10.1016/0021-9045(75)90080-5

Cited by 35 publications

(13 citation statements)

References 7 publications

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“…The proof of Theorem 2 and Theorem 3 is complete. □ If q = 1 (not covered by these theorems), two-sided ç-spline interpolation is essentially the same as cardinal spline interpolation, for which logarithmic growth of ||5|| with k has been demonstrated; see [6]. This fact supports the conjecture that q tending to zero gives "worst-case" results for the nodes (3.1).…”

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confidence: 56%

Spline Interpolation at Knot Averages on a Two-Sided Geometric Mesh

Marsden¹

1982

Mathematics of Computation

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Abstract. For splines of degree k > 1 with knots -ti, = t2m+i^¡ = 1 + q + q2 + ■ ■ ■ +qm~', i = 1, . . ., m, where 0 < q < 1, it is shown that spline interpolation to continuous functions at nodes t, = SÎ xvJl+j, i = 1, . . . , n = 2m -k -1, has operator norm H^H which is bounded independently of q and m as q tends to zero if and only if(1 -wtf <|, (1 -wk)k < j, and w, > 0, / ■■ 1,..., k. The choice of nodes t, = 2o + l wj'i+j arK* tne growth rate of ||P|| with k are also discussed.1. Two-Sided ¿^-Splines. To integers n > 0, k > 0, and a nondecreasing sequence t = (t¡)1+k+x with r, < ti+k+x, i = 1, ■ ■ ■ , n, is associated Sfc+ljt, the space of polynomial splines of order k + 1 with knot sequence t, defined by S¿+lit = span{TV,, . . ., Nn), where each N¡ = Nik+X is an appropriate normalized /»-spline.See [1] for specific details.

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confidence: 56%

Spline Interpolation at Knot Averages on a Two-Sided Geometric Mesh

Marsden¹

1982

Mathematics of Computation

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“…is proved by Richards [10]. The approximation property of spline interpolation easily follows from this lemma, viz.…”

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confidence: 88%

L p estimates of boundary integral equations for some nonlinear boundary value problems

Eggermont

Saranen

1990

Numer. Math.

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Summary.Recently, Galerkin and collocation methods have been analyzed for boundary integral equation formulations of some potential problems in the plane with nonlinear boundary conditions, and stability results and error estimates in the H1/2-norm have been proved (Ruotsalainen and Wendland, and Ruotsalainen and Saranen). We show that these results extend to L p setting without any extra conditions. These extensions are proved by studying the uniform boundedness of the inverses of the linearized integral operators, and then considering the nonlinear equations. The fact that in H 1/2 setting the nonlinear operator is a homeomorphism with Lipschitz continuous inverse plays a crucial role. Optimal error estimates for the Galerkin and collocation method in L p space then follow.

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“…В 1975 г. Ричардс [2] определил асимптотику констант Лебега для интер-поляционных кардинальных полиномиальных сплайнов степени r с узлами интерпо-ляции i ∈ Z и возможными разрывами r-й производной в узлах {i + (r + 1)/2, i ∈ Z}:…”

Section: математические заметкиunclassified

Точные константы Лебега для интерполяционных $\mathscr L$-сплайнов третьего порядка

Ким¹,

Kim²

2008

Mat. Zametki

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The Lebesgue constants for cardinal spline interpolation

Cited by 35 publications

References 7 publications

Spline Interpolation at Knot Averages on a Two-Sided Geometric Mesh

Spline Interpolation at Knot Averages on a Two-Sided Geometric Mesh

L p estimates of boundary integral equations for some nonlinear boundary value problems

Точные константы Лебега для интерполяционных $\mathscr L$-сплайнов третьего порядка

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