Abstract
Splines and quasi-interpolation operators are important both in approximation theory and applications. In this paper, we construct a family of quasi-interpolation operators for the bivariate quintic spline spaces S 5 3 ( Δ m n ( 2 ) ) . Moreover, the properties of the proposed quasi-interpolation operators are studied, as well as its applications for solving the two-dimensional Burgers’ equation and image reconstruction. Some numerical examples show that these methods, which are easy to implement, provide accurate results.
Highlights
Spline functions are very important in both approximation theory and applications in science and engineering
The special importance of spline functions is due to the mechanical meaning of the univariate spline, which was discussed in the famous paper written by Schoenberg [1]
We can see that the operator W1 has a better performance with smaller image distortion
Summary
Spline functions are very important in both approximation theory and applications in science and engineering. We construct bivariate quasi-interpolation operators to solve the 2D Burgers’ equation and reconstruct images. Since the cardinality of Gs1,t Ť Gs2,t is the same as the dimension of S53p∆pm2nqq, it is sufficient to prove that Gs1,t Ť Gs2,t is a linearly independent set on D This can be done by following the proof of Theorem 3.1 in [29]. The same fact can be obtained for I I, I I I, IV in Dij, respectively These theorems hold by the arbitrariness of Dij. we give two applications of the quasi-interpolation operator W1 in Theorem 7
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