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.     

On approximation of periodic functions by Fourier sums

Let L _p , 1 ≤ p< ∞, be the space of 2π-periodic functions f with the norm || f ||_p = ( ò _{- p}^p | f |^p )^{1 \mathord}

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.     

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.     

On the approximation of discrete functions by fourier-hahn sums

— f: {0,1, ..., N–1}R. .     

On feasible sets defined through Chebyshev approximation

LetZ be a compact set of the real space with at leastn + 2 points;f,h1,h2:Z continuous functions,h1,h2 strictly positive andP(x,z),x(x ₀,...,x _n ) ⁿ⁺¹,z , a polynomial of degree at mostn. Consider a feasible setM {x ⁿ⁺¹z Z, –h ₂(z) P(x, z)–f(z)h ₁(z)}. Here it is proved the null vector 0 of ⁿ⁺¹ belongs to the compact convex hull of the gradients ± (1,z,...,z ⁿ ), wherez Z are the index points in which the constraint functions are active for a givenx* M, if and only ifM is a singleton.This work was partially supported by CONACYT-MEXICO.     

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.     

Relative norms of Gauss sums

The explicit determination of the values of Gauss sums is a very classical problem and has some rather deep arithmetic consequences in classfield theory. Here we study the simpler problem of finding their relative norms. We give a complete determination of the relative norms of Gauss sums for norms whose values are known to be in an imaginary quadratic extension of the rational field.     

Chebyshev approximation to data by geometric elements

An algorithm is presented and proved correct, for the efficient approximation of finite point sets in ² and ³ by geometric elements such as circles, spheres and cylinders. It is shown that the approximation criterion used, viz. minimising the maximum orthogonal deviation, is best modelled mathematically through the concept of aparallel body. This notion, besides being a valuable tool for form assessment in metrology, contributes to approximation theory by introducing a new kind of approximation, here called geometric or orthogonal. This approach is closely related to but different from Chebyshev approximation.The work described is part of a Commission of the European Communities project (Contract 3327/1/0/158/89/9-BCR-UK(30)).     

On uniform approximation by partial sums of Jacobi series on elliptic region

In this paper, we discuss the relation between the partial sums of Jacobi series on an elliptic region and the corresponding partial sums of Fourier series. From this we derive a precise approximation formula by the partial sums of Jacobi series on an elliptic region.