On almost-everywhere convergence of Malmquist-Takenaka series

Almost Everywhere Convergence of Series in L

We answer positively a question of J. Rosenblatt (1988), proving the existence of a sequence (ci) with ∑∞ i=1 |ci| = ∞, such that for every dynamical system (X,Σ, m, T ) and f ∈ L1(X), ∑∞i=1 cif(T ix) converges almost everywhere. A similar result is obtained in the real variable context.

Mean and Almost Everywhere Convergence of Fourier-neumann Series

Let Jμ denote the Bessel function of order μ. The functions xJα+β+2n+1(x 1/2), n = 0, 1, 2, . . . , form an orthogonal system in L2((0,∞), xα+βdx) when α+ β > −1. In this paper we analyze the range of p, α and β for which the Fourier series with respect to this system converges in the Lp((0,∞), xαdx)-norm. Also, we describe the space in which the span of the system is dense and we show some of ...

The surprising almost everywhere convergence of Fourier-Neumann series

For most orthogonal systems and their corresponding Fourier series, the study of the almost everywhere convergence for functions in L requires very complicated research, harder than in the case of the mean convergence. For instance, for trigonometric series, the almost everywhere convergence for functions in L is the celebrated Carleson theorem, proved in 1966 (and extended to L by Hunt in 1967...

Fourier analysis and the Takenaka-Malmquist basis

We consider problems related to the generalization of the classical Fourier basis to a basis of rational functions with prescribed poles outside the unit disk. We give some generalizations about the convergence and estimation of the Fourier coefficients with respect to this generalized basis. We also consider a rational generalization of the classical Toeplitz matrices and consider the asymptot...

Almost Everywhere Convergence of Poisson Integrals on Generalized Half-planes