Lidstone approximation on the triangle

Costabile, F.; Dell’Accio, Francesco

doi:10.1016/j.apnum.2004.08.003

Cited by 25 publications

(16 citation statements)

References 15 publications

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“…Here we integrate on the standard simplex the Lidstone interpolation polynomial P L M f (x, y) [8], which reproduces exactly the polynomials of P 2M−1 (x,y) . This polynomial can be rewritten in terms of a basis which involves the Lidstone polynomials n (x) [2], with scalar coefficients which are even order mixed derivatives involving ∂/∂ x, ∂/∂y and the directional derivative…”

Section: Strongly Embedded Btqfs For the Simplex With Adementioning

confidence: 97%

“…In the papers [7,8,11] three polynomial expansions like (2) for smooth enough functions on the simplex are proposed, with the following common features:…”

Section: Strongly Embedded Btqfs For the Simplex With Adementioning

confidence: 99%

“…The paper is organized as follows. According to the proposed strategy, in Section 2 we get formulas of type (3) by integrating some polynomial expansions for smooth enough functions on 2 : the expansion in Bernoulli polynomials [7], the Lidstone expansion [8] and the Lidstone expansion of 2nd type [11], that generalize to the simplex the corresponding above mentioned one dimensional expansions. Bounds for the remainder and sufficient conditions for convergence are explicitly given in the case of Lidstone quadrature; in the remaining cases similar conditions hold.…”

Section: Introductionmentioning

confidence: 99%

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New embedded boundary-type quadrature formulas for the simplex

Costabile

Dell’Accio

2007

Numer Algor

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In this paper we consider the problem of the approximation of the integral of a smooth enough function f (x, y) on the standard simplex 2 ⊂ IR 2 by cubature formulas of the following kind:where the nodes (x α , y α ) , α = 1, 2, 3 are the vertices of the simplex. Such kind of quadratures belong to a more general class of formulas for numerical integration, which are called boundary-type quadrature formulas. We discuss three classes of such formulas that are exact for algebraic polynomials and generate embedded pairs. We give bounds for the truncation errors and conditions for convergence. Finally, we show how to organize an algorithm for the automatic computation of the quadratures with estimate of the errors and provide some numerical examples.Keywords Boundary type quadrature · Simplex · Algebraic degree of exactness Mathematics Subject Classifications (2000)Primary 65D30 · 65D32 · Secondary 65D05 · 65D15 To Prof. Walter Gautschi for his 50 years of professional activity.

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Section: Strongly Embedded Btqfs For the Simplex With Adementioning

confidence: 97%

“…In the papers [7,8,11] three polynomial expansions like (2) for smooth enough functions on the simplex are proposed, with the following common features:…”

Section: Strongly Embedded Btqfs For the Simplex With Adementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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New embedded boundary-type quadrature formulas for the simplex

Costabile

Dell’Accio

2007

Numer Algor

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“…respectively, we proceed by adapting, in this general case, the extension technique to the simplex specially developed in [13,14] for expansions in Bernoulli and Lidstone polynomials (see also [18] for extensions of asymmetric expansions). We denote by r 1 the line through V 2 , V 3 , by r 2 the line through V 1 , V 3 and by r 3 the line through…”

Section: 1mentioning

confidence: 99%

Bivariate Shepard–Bernoulli operators

Dell’Accio

Tommaso

2017

Mathematics and Computers in Simulation

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Abstract. In this paper we extend the Shepard-Bernoulli operators introduced in [6] to the bivariate case. These new interpolation operators are realized by using local support basis functions introduced in [23] instead of classical Shepard basis functions and the bivariate three point extension [13] of the generalized Taylor polynomial introduced by F. Costabile in [11]. The new operators do not require either the use of special partitions of the node convex hull or special structured data as in [8]. We deeply study their approximation properties and provide an application to the scattered data interpolation problem; the numerical results show that this new approach is comparable with the other well known bivariate schemes QSHEP2D and CSHEP2D by Renka [34,35]. The problemLet N = {x 1 , ..., x N } be a set of N distinct points (called nodes or sample points) of R s , s ∈ N, and let f be a function defined on a domain D containing N . The classical Shepard operators (first introduced in [37] in the particular case s = 2) are defined bywhere the weight functions A µ,i (x) in barycentric form areand |·| denotes the Euclidean norm in R s . The interpolation operator S N,µ [·] is stable, in the sense thatbut for µ > 1 the interpolating function S N,µ [f ] (x) has flat spots in the neighborhood of all data points. Moreover, the degree of exactness of the operator S N,µ [·] is 0, i.e. if it is restricted to the polynomial space P

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“…Further characterization can be found in the study of [9][10][11][12][13][14][15][16]. More research on Lidstone interpolation as well as Lidstone spline is seen in [1,[17][18][19][20][21][22][23]. On the other hand, the Lidstone boundary value problems and several of its particular cases have been the subject matter of numerous investigations, see [4,18,[24][25][26][27][28][29][30][31][32][33][34][35][36][37] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Eigenvalues of complementary Lidstone boundary value problems

Agarwal

Wong

2012

Bound Value Probl

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We consider the following complementary Lidstone boundary value problemwhere l > 0. The values of l are characterized so that the boundary value problem has a positive solution. Moreover, we derive explicit intervals of l such that for any l in the interval, the existence of a positive solution of the boundary value problem is guaranteed. Some examples are also included to illustrate the results obtained. Note that the nonlinear term F depends on y' and this derivative dependence is seldom investigated in the literature.

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Lidstone approximation on the triangle

Cited by 25 publications

References 15 publications

New embedded boundary-type quadrature formulas for the simplex

New embedded boundary-type quadrature formulas for the simplex

Bivariate Shepard–Bernoulli operators

Eigenvalues of complementary Lidstone boundary value problems

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