On Location and Approximation of Clusters of Zeros of Analytic Functions

At the beginning of the 1980s, M. Shub and S. Smale developed a quantitative analysis of Newton's method for multivariate analytic maps. In particular, their α-theory gives an effective criterion that ensures safe convergence to a simple isolated zero. This criterion requires only information concerning the map at the initial point of the iteration. Generalizing this theory to multiple zeros and clusters of zeros is still a challenging problem. In this paper we focus on one complex variable function. We study general criteria for detecting clusters and analyze the convergence of Schroder's iteration to a cluster. In the case of a multiple root, it is well known that this convergence is quadratic. In the case of a cluster with positive diameter, the convergence is still quadratic provided the iteration is stopped sufficiently early. We propose a criterion for stopping this iteration at a distance from the cluster which is of the order of its diameter.     

Spectrum is periodic for n-intervals

In this paper we study spectral sets which are unions of finitely many intervals in R. We show that any spectrum associated with such a spectral set Ω is periodic, with the period an integral multiple of the measure of Ω. As a consequence we get a structure theorem for such spectral sets and observe that the generic case is that of the equal interval case.     

A note on the improved derivative free root-solvers

Using a fixed point relation of the square-root type and the basic fourth-order method, improved methods of fifth and sixth order for the simultaneous determination of simple zeros of a polynomial are obtained. An increase in convergence is achieved without additional numerical operations, which points to high computational efficiency of the accelerated methods. The main aim of this work is the convergence analysis of improved simultaneous methods given under computationally verifiable initial conditions in the spirit of Smale’s point estimation theory.     

Orthonormal polynomials with exponential-type weights

Let and let w_ρ(x)|x|^ρexp(-Q(x)), where and is an even function. In this paper we consider the properties of the orthonormal polynomials with respect to the weight , obtaining bounds on the orthonormal polynomials and spacing on their zeros. Moreover, we estimate A_n(x) and B_n(x) defined in Section 4, which are used in representing the derivative of the orthonormal polynomials with respect to the weight .     

A companion matrix approach to the study of zeros and critical points of a polynomial

In this paper, we introduce a new type of companion matrices, namely, D-companion matrices. By using these D-companion matrices, we are able to apply matrix theory directly to study the geometrical relation between the zeros and critical points of a polynomial. In fact, this new approach will allow us to prove quite a number of new as well as known results on this topic. For example, we prove some results on the majorization of the critical points of a polynomial by its zeros. In particular, we give a different proof of a recent result of Gerhard Schmeisser on this topic. The same method allows us to prove a higher order Schoenberg-type conjecture proposed by M.G. de Bruin and A. Sharma.     

On the zeros of basic finite Hankel transforms

In this article we prove that the basic finite Hankel transform whose kernel is the third-type Jackson q-Bessel function has only infinitely many real and simple zeros, provided that q satisfies a condition additional to the standard one. We also study the asymptotic behavior of the zeros. The obtained results are applied to investigate the zeros of q-Bessel functions as well as the zeros of q-trigonometric functions. A basic analog of a theorem of G. Pólya (1918) on the zeros of sine and cosine transformations is also given.     

Asymptotics of zeros of polynomials arising from rational integrals

We prove that the zeros of the polynomials Pm(a) of degree m, defined by Boros and Moll via

     

Some Indeterminate Moment Problems and Freud-Like Weights

We study two indeterminate Hamburger moment problems and the corresponding orthogonal polynomials. The coefficients in their recurrence relations are of exponential growth or are polynomials of degree 2. The entire functions in the Nevanlinna parametrization are found. The orthogonal polynomials with polynomial recurrence coefficients resemble the Freud polynomials with a = 1/2 . Inequalities are given for the largest zero and the asymptotic behavior of the largest zero is established. April 24, 1996. Date revised: March 3, 1997.     

Existence theorems for m-accretive operators in Banach spaces

In 1985, the second author proved a surjective result for m-accretive and ?-expansive mappings for uniformly smooth Banach spaces. However, in this case, we have been able to remove the uniform smoothness of the Banach space, without any additional assumption.     

Some inequalities for symmetric functions and an application to orthogonal polynomials

We present some sharp inequalities for symmetric functions and give an application to orthogonal polynomials.