Juan L. Varona

Graphic and numerical comparison between iterative methods

generates a sequence {xn}n=0 that converges to ζ. In fact, Newton’s original ideas on the subject, around 1669, were considerably more complicated. A systematic study and a simplified version of the method are due to Raphson in 1690, so this iteration scheme is also known as the Newton-Raphson method. (Also as the tangent method, from its geometric interpretation.) In 1879, Cayley tried to use the method to find complex roots of complex functions f : C → C. If we take z0 ∈ C and we iterate

Asymptotic estimates for Apostol-Bernoulli and Apostol-Euler polynomials

We analyze the asymptotic behavior of the Apostol-Bernoulli polynomials

The “Gauss-Seidelization” of iterative methods for solving nonlinear equations in the complex plane

\mathcal{B}_{n}(x;\lambda)

Fourier series of functions whose Hankel transform is supported on [0, 1]

in detail. The starting point is their Fourier series on

Two notes on convergence and divergence a.e. of Fourier series with respect to some orthogonal systems

[0,1]

Asymptotic behaviour of orthogonal polynomials relative to measures with mass points

which, it is shown, remains valid as an asymptotic expansion over compact subsets of the complex plane. This is used to determine explicit estimates on the constants in the approximation, and also to analyze oscillatory phenomena which arise in certain cases. These results are transferred to the Apostol-Euler polynomials

Harmonic analysis associated with a discrete Laplacian

\mathcal{E}_{n}(x;\lambda)

The Möbius inversion formula for Fourier series applied to Bernoulli and Euler polynomials

via a simple relation linking them to the Apostol-Bernoulli polynomials.

Some asymptotic properties for orthogonal polynomials with respect to varying measures

Abstract In this paper we introduce a process we have called “Gauss-Seidelization” for solving nonlinear equations. We have used this name because the process is inspired by the well-known Gauss–Seidel method to numerically solve a system of linear equations. Together with some convergence results, we present several numerical experiments in order to emphasize how the Gauss-Seidelization process influences on the dynamical behavior of an iterative method for solving nonlinear equations.

A Simple Computation of ζ (2 k )

LetJμ denote the Bessel function of order μ. For α>−1, the system x−α/2−1/2Jα+2n+1(x1/2, n=0, 1, 2,..., is orthogonal onL2((0, ∞),xαdx). In this paper we study the mean convergence of Fourier series with respect to this system for functions whose Hankel transform is supported on [0, 1].