Chebyshev Multivariate Polynomial Approximation and Point Reduction Procedure
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We apply the methods of nonsmooth and convex analysis to extend the study of Chebyshev (uniform) approximation for univariate polynomial functions to the case of general multivariate functions (not just polynomials). First of all, we give new necessary and sufficient optimality conditions for multivariate approximation, and a geometrical interpretation of them which reduces to the classical alternating sequence condition in the univariate case. Then, we present a procedure for verification of necessary and sufficient optimality conditions that is based on our generalization of the notion of alternating sequence to the case of multivariate polynomials. Finally, we develop an algorithm for fast verification of necessary optimality conditions in the multivariate polynomial case.
KeywordsMultivariate polynomials Chebyshev approximation Best approximation conditions
Mathematics Subject Classification49J52 90C26 41A15 41A50
This research was supported by the Australian Research Council (ARC), Solving hard Chebyshev approximation problems through nonsmooth analysis (Discovery Project DP180100602). This paper was inspired by the discussions during the program “Approximation and Optimization” which took place in July 2016, at the residential mathematical research institute MATRIX in Creswick, Australia. We are thankful to the MATRIX organizers, support team and participants for a terrific research atmosphere and productive discussions. Finally, we would like to thank the referee for his/her valuable and constructive advice and comments.
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