Integro spline quasi-interpolants and their super convergence
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Integro spline quasi-interpolants and their super convergence Jinming Wu1 · Wurong Ge1 · Xiaolei Zhang1 Received: 10 March 2020 / Revised: 7 July 2020 / Accepted: 18 July 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract This paper gives a type of new spline quasi-interpolants where the entry values are integral values of successive intervals, rather than the usual function values at the knots. They are called integro spline quasi-interpolants. Also, their super convergence property in approximating function values/derivative values at the knots/mid-knots are proved. Numerical experiments show that the integro spline quasi-interpolants possess super convergence. Keywords Spline quasi-interpolants · Integral values · Super convergence
1 Introduction Assume that f (x) is a unknown univariate function and let X N := {a = x 0 < x 1 < · · · < x N −1 < x N = b}
be the uniform partition of [a, b] with the step length h = (b − a)/N . In the traditional interpolation problem, the function values at the knots are known. But now we deal with the situations which only involve the integral values over the subintervals and denote by Ii ( f ) these integrals (when no ambiguity occurs, we abbreviate Ii ( f ) as Ii for simplicity), i.e., xi+1
f (x)d x = Ii , i = 0, 1, · · · , N − 1.
(1)
xi
This problem arises frequently in many fields, such as numerical analysis, mathematical statistics, environmental science, climatology, oceanography and so on Epstein (1991); Killworth (1996); Delhez (2003); Huang and Chen (2005); Moghaddam et al. (2017). It is
Communicated by Cassio Oishi. This work was supported by the Natural Science Foundation of Zhejiang Province (Grant No. LY19A010003) and the National Natural Science Foundation of China (Grant No. 11671068).
B 1
Jinming Wu [email protected] School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China 0123456789().: V,-vol
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an interesting question and it has received some attentions in the past 15 years. In 2006, Behforooz (2006) first introduced a method to construct integro cubic splines by using the given integral values over successive intervals, rather than the usual function values at the knots. Subsequently, some researchers (Behforooz (2010), Zhanlav and Mijiddorj (2010), Zhanlav and Mijiddorj (2017), Lang and Xu (2012), Lang and Xu (2018), Wu and Zhang (2013, 2015)) are devoted to working on this issue. In general, spline functions can be successfully applied to solve this problem and they possess good approximation behaviors. However, all the existing methods are required to solve a system of linear equations and some additional boundary conditions are demanded. Spline quasi-interpolant is an effective and practical approximation operator. It has many advantages such as good shape preserving property, polynomial reproduction property, easy computation and evaluation, and so on. Especially, Sablonniere (2005) presented discrete univariate spline quasi-interpola
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