Chebyshev Multivariate Polynomial Approximation: Alternance Interpretation
In this paper, we derive optimality conditions for Chebyshev approximation of multivariate functions. The theory of Chebyshev (uniform) approximation for univariate functions was developed in the late nineteenth and twentieth century. The optimality conditions are based on the notion of alternance (maximal deviation points with alternating deviation signs). It is not clear, however, how to extend the notion of alternance to the case of multivariate functions. There have been several attempts to extend the theory of Chebyshev approximation to the case of multivariate functions. We propose an alternative approach, which is based on the notion of convexity and nonsmooth analysis.
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This paper was inspired by the discussions during a recent MATRIX program “Approximation and Optimisation’’ that took place in July 2016. We are thankful to the MATRIX organisers, support team and participants for a terrific research atmosphere and productive discussions.
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