We present algorithms for the determination of polynomials orthogonal with respect to a positive weight function multiplied by a polynomial with simple roots inside the interval of integration. We apply these algorithms to search for and calculate all possible sequences of imbedded quadratures of maximal polynomials order of precision for the generalized Laguerre and Hermite weight functions.
In the present paper, we prove a global direct theorem for the modified Baskakov type operators in terms of so called Ditzian-Totik modulus of smoothness.
We study nonlinear m-term approximation in a Banach space with regard to a basis. It is known that in the case of a greedy basis (like the Haar basis in , 1 < p < ∞) a greedy type algorithm realizes nearly best m-term approximation for any individual function. In this paper we generalize this result in two directions. First, instead of a greedy algorithm we consider a weak greedy algorithm. Second, we study in detail unconditional nongreedy bases (like the multivariate Haar basis in ,...
*This research was supported by the National Science Foundation Grant DMS 0200187 and by ONR Grant N00014-96-1-1003This paper is a survey which also contains some new results on the nonlinear approximation with regard to a basis or, more generally, with regard to a minimal system. Approximation takes place in a Banach or in a quasi-Banach space. The last decade was very successful in studying nonlinear approximation. This was motivated by numerous applications. Nonlinear approximation is important...