Viewing the classical Bernstein polynomials as sampling operators, we study a generalization by allowing the sampling operation to take place at scattered sites. We utilize both stochastic and deterministic approaches. On the stochastic side, we consider the sampling sites as random variables that obey some naturally derived probabilistic distributions, and obtain Chebyshev type estimates. On t...

‎It is commonly accepted that fractional differential equations play‎ ‎an important role in the explanation of many physical phenomena‎. ‎For‎ ‎this reason we need a reliable and efficient technique for the‎ ‎solution of fractional differential equations‎. ‎This paper deals with‎ ‎the numerical solution of a class of fractional differential‎ ‎equation‎. ‎The fractional derivatives are described...

The concept of H-sets as introduced by Collatz in 1956 was very useful univariate Chebyshev approximation polynomials or spaces. In the multivariate setting, situation is much worse, because there no alternation, and exist, but are only rarely accessible mathematical arguments. However, Reproducing Kernel Hilbert spaces, shown here to have a rather simple complete characterization. As byproduct...

In this article it is shown how to improve the numerical stability of a discrete least-squares method for computing eigenvalue approximations of the Laplace operator defined on standard two-dimensional domains. Among many sets of matching points the smallest condition numbers of the corresponding matrices have been obtained by using the Morrow-Patterson and Padua points in the square case, the ...

We evaluate the matrix elements 〈Orp〉, where O = {1, β, iαnβ} are the standard Dirac matrix operators and the angular brackets denote the quantum-mechanical average for the relativistic Coulomb problem, in terms of generalized hypergeometric functions 3F2 (1) for all suitable powers. Their connections with the Chebyshev and Hahn polynomials of a discrete variable are emphasized. As a result, we...

Spherical r-designs are Chebyshev-type averaging sets on the d-dimensional unit sphere Sd-l that are exact for all polynomials of degree at most t. The concept of such designs was introduced by Delsarte , Goethals and Seidel in 1977. The existence of spherical designs for every t and d was proved by Seymour and Zaslavsky in 1984. Although some sporadic examples are known, no general constructio...

Laurent polynomials related to the Hahn-Exton q-Bessel function, which are qanalogues of the Lommel polynomials, have been introduced by Koelink and Swarttouw. The explicit strong moment functional with respect to which the Laurent q-Lommel polynomials are orthogonal is given. The strong moment functional gives rise to two positive definite moment functionals. For the corresponding sets of orth...

Amongst satisfactory techniques for the numerical solution of differential equations, the use of Chebyshev series is often avoided because of the tedious nature of the calculations. A systematic application of the Chebyshev method is given for certain fourth order boundary value problems in which the derivatives have polynomial coefficients. Numerical results for various problems using the Cheb...

We consider Chebyshev polynomials, Tn(z), for infinite, compact sets e ⊂ R (that is, the monic polynomials minimizing the sup–norm, ‖Tn‖e, on e). We resolve a 45+ year old conjecture of Widom that for finite gap subsets of R, his conjectured asymptotics (which we call Szegő–Widom asymptotics) holds. We also prove the first upper bounds of the form ‖Tn‖e ≤ QC(e) (where C(e) is the logarithmic ca...

We consider Chebyshev polynomials, Tn(z), for infinite, compact sets e ⊂ R (that is, the monic polynomials minimizing the sup–norm, ‖Tn‖e, on e). We resolve a 45+ year old conjecture of Widom that for finite gap subsets of R, his conjectured asymptotics (which we call Szegő–Widom asymptotics) holds. We also prove the first upper bounds of the form ‖Tn‖e ≤ QC(e) (where C(e) is the logarithmic ca...