Univariate Lidstone-type multiquadric quasi-interpolants
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Univariate Lidstone-type multiquadric quasi-interpolants Ruifeng Wu1,3 · Huilai Li2 · Tieru Wu2,3 Received: 18 March 2018 / Revised: 11 June 2019 / Accepted: 8 April 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, a kind of univariate multiquadric quasi-interpolants with the derivatives of approximated function is proposed by combining a univariate multiquadric quasi-interpolant with Lidstone interpolation polynomials proposed in Lidstone (Proc Edinb Math Soc 2:16– 19, 1929), Costabile and Dell’ Accio (App Numer Math 52:339–361, 2005) and Catinas (J Appl Funct Anal 4:425–439, 2006). For practical purposes, another kind of approximation operators without any derivative of the approximated function is given using divided differences to approximate the derivatives. Some error bounds and the convergence rates of new operators are derived, which demonstrates that our operators could provide the desired precision by choosing a suitable shape-preserving parameter c and a non-negative integer n. Finally, we make extensive comparison with the other existing methods and give some numerical examples. Moreover, the associated algorithm is easily implemented. Keywords Multiquadrics · Quasi-interpolants · Lidstone interpolation polynomials · Convergence rates Mathematics Subject Classification 41A05 · 65D05 · 65D15
Communicated by Antonio José Silva Neto.
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Tieru Wu [email protected] Ruifeng Wu [email protected] Huilai Li [email protected]
1
School of Applied Mathematics, Jilin University of Finance and Economics, Changchun 130117, People’s Republic of China
2
School of Mathematics, Jilin University, Changchun 130012, People’s Republic of China
3
Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun 130012, People’s Republic of China 0123456789().: V,-vol
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1 Introduction Let X = {x0 , . . . , x N } be a set of N + 1 distinct points of R and let f be a function defined on a domain [a, b] containing X . The standard formula for interpolating the function f , where: a = x0 < · · · < x N = b,
(1)
has the following form: L[ f ; a, b](x) =
N
λ j X (x − x j ),
(2)
j=0
show that L[ f ; a, b](x j ) = f (x j ),
(3)
for all j = 0, 1, . . . , N , where X (·) is an interpolation kernel. Many investigators use radial basis functions to solve the interpolation problem (2)–(3). In particular, the multiquadrics presented by Hardy (1971): φ j (x) = φ(x − x j ) = (x − x j )2 + c2 , j = 0, 1, . . . , N , (4) are of especial interest because of their special convergence property [see (Buhmann 1990a, b)]. Throughout this paper, let the notations φ j (·) and c denote the multiquadrics and their shape-preserving parameter as in (4), respectively. A review by Fkanke (1982) indicated that the multiquadric interpolation is one of the best schemes among some 29 interpolation methods in terms of accuracy, efficiency, and easy implementation. Although the multiquadric inter
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