The Methods And Applications Of Constructing High-precision Quasi-interpolation Based On Bernstein Quasi-interpolation Scheme

The celebrated Weierstrass Approximation Theorem (1885) heralded intermittent interest in polynomial approximation, which continues unabated even as of today. The Russian mathematician Bernstein in 1912 not only provided an interesting proof of the Weierstrass theorem, but also displayed a sequence of polynomials that approximate a given function f∈C[0,1]. Due to Bernstein polynomials are widely applied in computer-aided geometric design (CAGD), approximation theory and financial mathe-matics, until today, this research approach is still a hot spot of approximation theorem research.This article mainly discusses how to make full use of the successive function of Bernstein quasi-interpolation operator and quasi-inverse in polynomial space to find a more accurate approximation, namely using Boolean sum of Bernstein quasi-interpolation and left-inverse operator of Bernstein operator in polynomial space con-structed high-order quasi-interpolation. During my research, I find the best approxima-tion number and then give the best approximation of such approximation.Finally, the research verifies the application of Bernstein operator'Boolean sum by the actual numerical example and approximation convergence through iteration con-structing, and also gives reasonable relationship between the number of sampling points and iteration steps when the approximation order is in its best.