Nonnegative trigonometric polynomials

An extremal problem for the coefficients of sine polynomials, which are nonnegative in [0,π] , posed and discussed by Rogosinski and Szego is under consideration. An analog of the Fejér-Riesz representation of nonnegative general trigonometric and cosine polynomials is proved for nonnegative sine polynomials. Various extremal sine polynomials for the problem of Rogosinski and Szego are obtained explicitly. Associated cosine polynomials k n (θ) are constructed in such a way that { k n (θ) } are summability kernels. Thus, the L p , pointwise and almost everywhere convergence of the corresponding convolutions, is established. © 2002 Springer-Verlag New York Inc.

An extremal nonnegative sine polynomial

For any positive integer n, the sine polynomials that are nonnegative in [0, π] and which have the maximal derivative at the origin are determined in an explicit form. Associated cosine polynomials Kn (θ) are constructed in such a way that {Kn(θ)} is a summability kernel. Thus, for each Pi 1 ≤ P ≤ ∞ and for any 27π-periodic function f ∈ Lp [-π, π], the sequence of convolutions Kn * f is proved to converge to f in Lp[-ππ]. The pointwise and almost everywhere convergences are also consequences of our construction.

An environment for knowledge discovery in biology

This paper describes a data mining environment for knowledge discovery in bioinformatics applications. The system has a generic kernel that implements the mining functions to be applied to input primary databases, with a warehouse architecture, of biomedical information. Both supervised and unsupervised classification can be implemented within the kernel and applied to data extracted from the primary database, with the results being suitably stored in a complex object database for knowledge discovery. The kernel also includes a specific high-performance library that allows designing and applying the mining functions in parallel machines. The experimental results obtained by the application of the kernel functions are reported. © 2003 Elsevier Ltd. All rights reserved.