Approximation by interpolation operators with piecewise polynomial basis functions

Abstract

This paper focuses on approximation by interpolation operators with piecewise polynomial basis functions. It first shows that the interpolation operator introduced by Zhan, Huang and Liu ( A class of interpolation basis functions with excellent properties, science in China , Mathematics 53 (2023), 1325–1348) can be interpreted as a neural network activated by piecewise polynomials, explicitly describing its neural network topology. Then, it proves that the approximation rate of this operator for continuous functions can be characterized by the third-order modulus of smoothness of the target function, improving the results in (R. J. Zhang, Z. P. Huang, and X. Liu, A class of interpolation basis functions with excellent properties, science in China , Mathematics 53 (2023), 1325–1348), and the obtained approximation rate estimate is best possible. A saturation theorem for the approximation by this operator is also presented, showing that the norm of the difference between the operator and the function is of order o (1/ n ³ ) if and only if the function is a polynomial of degree at most 2. Additionally, a new interpolation operator with derivative information is introduced, establishing sharp approximation rates as well as simultaneous approximation rates for smooth functions. Finally, a quasi-interpolation operator is proposed, whose approximation rate can be estimated by the modulus of smoothness of order r + 1. Numerical experiments are presented to investigate the influence of the order parameter, compare the proposed constructions with representative classical and neural-network interpolation methods, and assess their stability and computational behavior.