GENERALIZED SIMULTANEOUS CHEBYSHEV APPROXIMATION

In this note,we develop,without assuming the Haar condition,a generalized simultaneousChebyshev approximation theory which is similar to the classical Chebyshev theory and con-rains it as a special case.Our results also contain those in[1]and[3]as a special case,and thetwo conjectures proposed by C.B.Dunham in[2]are proved to be true in the case of simulta-neous approximation.     

Weighted discrete nonlinear Chebyshev approximation

Modifications are given to let the discrete REMEZ program for nnlinear Chebyshev approximation handle weights.     

On nonlinear simultaneous Chebyshev approximation problems

This paper is concerned with the problem of nonlinear simultaneous Chebyshev approximation in a real continuous function space. Some results on existence are established, in addition to characterization conditions of Kolmogorov type and also of alternation type. Applications are given to approximation by rational functions, by exponential sums and by Chebyshev splines with free knots.     

Weighted simultaneous approximation with Baskakov type operators

We obtain inequalities for the weighted approximation error of Baskakov type operators and their derivatives. Such inequalities are valid for functions of polynomial growth and are expressed in terms of weighted moduli of continuity.     

Linear Chebyshev approximation without Chebyshev sets

The provision of algorithms for computing best Chebyshev approximations on a continuum by general linear combinations of continuous functions is considered. Four possible approaches are described, and detailed comparisons are given for some test problems.AMS (MOS) subject classifications (1970). Primary 65D10; Secondary 41A50.     

Chebyshev approximation by products

Characterization and uniqueness of minimax approximation by the product PQ of two finite dimensional subspaces P and Q is studied. Some approximants may have no standard characterization since PQ may not be a sun, but interior points do have the standard linear characterization.     

Adaptive bivariate Chebyshev approximation

We propose an adaptive algorithm which extends Chebyshev series approximation to bivariate functions, on domains which are smooth transformations of a square. The method is tested on functions with different degrees of regularity and on domains with various geometries. We show also an application to the fast evaluation of linear and nonlinear bivariate integral transforms. Work supported by the research project CPDA028291 “Efficient approximation methods for nonlocal discrete transforms” of the University of Padova, and by the GNCS-INdAM.     

Chebyshev approximation by exponential-polynomial sums

It is shown that best Chebyshev approximations by exponential-polynomial sums are characterized by (a variable number of) alternations of their error curve and are unique. Computation of best approximations via the Remez algorithm and Barrodale approach is considered.     

The discrete linear restricted Chebyshev approximation

A numerically stable simplex algorithm for calculating the restricted Chebyshev solution of overdetermined systems of linear equations is described. In this algorithm minimum computer storage is required and no conditions are imposed on the coefficient matrix or on the right hand side of the system of equations. Also a new way of implementing a triangular decomposition method to the basis matrix is used. The ordinary Chebyshev solution, the one-sided Chebyshev solutions and the Chebyshev approximation by non-negative functions are obtained as special cases in this algorithm. Numerical results are given.