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If a closed subset of a Riemann surface is a set of uniform meromorphic approximation, then its boundary is shown to be a set of tangential meromorphic approximation. [ABSTRACT FROM AUTHOR]

For k ≥ 2 even, and α ≥ − ( 2 k + 1 ) ∕ 4 , we provide a uniform approximation of the ultraspherical polynomials P k ( α , α ) ( x ) in the oscillatory region with a very explicit error term. In fact, our result covers all α for which the expression “oscillatory region” makes sense. To that end, we construct the almost equioscillating function g ( x ) = c b ( x ) ( 1 − x 2 ) ( α + 1 ) ∕ 2 P k ( α , α ) ( x ) = cos B ( x ) + r ( x ) . Here the constant c = c ( k , α ) is defined by the normalization of P k ( α , α ) ( x ) , B ( x ) = ∫ 0 x b ( x ) d x , and the functions b ( x ) and B ( x ) , as well as bounds on the error term r ( x ) , are given by some rather simple elementary functions. [ABSTRACT FROM AUTHOR]

In this article we study questions related to the approximation of Markov functions on a finite set of points on the real line by rational functions with real coefficients. The main results include formulas for the error in best rational approximation. A connection of rational approximation problems with the discrete Hankel operator is also investigated. [ABSTRACT FROM AUTHOR]

Abstract: Consider -hulls, , from a dictionary of functions in space for . Their precise metric entropy orders are derived. Sparse linear approximation bounds are obtained to characterize the number of terms needed to achieve accurate approximation of the best function in a -hull that is closest to a target function. Furthermore, in the special case of , it is shown that a weak orthogonal greedy algorithm achieves the optimal approximation under an additional condition. [Copyright &y& Elsevier]

Kernel based methods provide a way to reconstruct potentially high-dimensional functions from meshfree samples, i.e., sampling points and corresponding target values. A crucial ingredient for this to be successful is the distribution of the sampling points. Since the computation of an optimal selection of sampling points may be an infeasible task, one promising option is to use greedy methods. Although these methods may be very effective, depending on the specific greedy criterion the chosen points might quickly lead to instabilities in the computation. To circumvent this problem, we introduce and investigate a new class of stabilized greedy kernel algorithms, which can be used to create a scale of new selection strategies. We analyze these algorithms, and in particular we prove convergence results and quantify in a precise way the distribution of the selected points. These results allow to prove, in the case of certain Sobolev kernels, that the algorithms have optimal stability and optimal convergence rates, including for functions outside the native space of the kernel. The results also apply to the case of the usual P -greedy algorithm, significantly improving state-of-the-art results available in the literature. Illustrative experiments are presented that support the theoretical findings and show improvements of the stabilized algorithms in terms of accuracy due to improved stability. [ABSTRACT FROM AUTHOR]

Results on two different settings of asymptotic behavior of approximation characteristics of individual functions are presented. First, we discuss the following classical question for sparse approximation. Is it true that for any individual function from a given function class its sequence of errors of best sparse approximations with respect to a given dictionary decays faster than the corresponding supremum over the function class? Second, we discuss sparse approximation by greedy type algorithms. We show that for any individual function from a given class we can improve the upper bound on the rate of convergence of the error of approximation by a greedy algorithm if we use some information from the previous iterations of the algorithm. Bounds of this type fall in the category of a posteriori bounds. [ABSTRACT FROM AUTHOR]