We propose a restarted Arnoldi's method with Faber polynomials and discuss its use for computing the rightmost eigenvalues of large non hermitian matrices. We illustrate, with the help of some practical test problems, the beneet obtained from the Faber acceleration by comparing this method with the Chebyshev based acceleration. M ethode d'Arnoldi-Faber pour les probl emes aux valeurs propres no...

For a certain subclass of Bazilevic functions, Faber polynomials expansions are used to obtain bi-univalent properties. Estimates on the $n$th Taylor-Maclaurin coefficients functions in this class found. Moreover, some special cases also indicated.

Given an analytic Jordan curve with interior G and exterior and given a sequence of complex numbers fang n satisfying lim sup n janj n we consider here three series of the form f z X n anPn z where the polynomials Pn z are chosen to be i the Faber polynomials associated with G ii the polynomials orthogonal over the area of G and iii the polynomials orthogonal over the contour Here we study the ...

Let S ⊂ R be compact with #S = ∞ and let C(S) be the set of all real continuous functions on S. We ask for an algebraic polynomial sequence (Pn) ∞ n=0 with deg Pn = n such that every f ∈ C(S) has a unique representation f = ∑∞ i=0 αiPi and call such a basis Faber basis. In the special case of S = Sq = {qk; k ∈ N0} ∪ {0}, 0 < q < 1, we prove the existence of such a basis. A special orthonormal F...

We consider five plots of zeros corresponding to four eponymous planar polynomials (Szegő, Bergman, Faber and OPUC), for degrees up to 60, and state five conjectures suggested by these plots regarding their asymptotic distribution of zeros. By using recent results on zero distribution of polynomials we show that all these “natural” conjectures are false. Our main purpose is to provide the theor...

The support of the orthogonality measure of so-called little q-Laguerre polynomials {ln(.; a|q)}n=0, 0 < q < 1, 0 < a < q−1, is given by Sq = {1, q, q, . . .} ∪ {0}. Based on a method of MÃlotkowski and Szwarc we deduce a parameter set which admits nonnegative linearization. We additionally use this result to prove that little q-Laguerre polynomials constitute a so-called Faber basis in C(Sq).