In this paper, we give characterizations of Riesz bases and near Riesz bases in Banach spaces. The notion of atomic system is defined and a characterization of atomic system has been given. Also results exhibiting relationship between frames, atomic systems and Riesz bases have been proved. Further, we show that every atomic system is a projection of a Riesz basis in Banach spaces. Finally, we ...

Generalized Riesz products similar to the type which arise as the spectral measure for a rank-one transformation are studied. A condition for the mutual singularity of two such measures is given. As an application, a probability space of transformations is presented in which almost all transformations are singular with respect to Lebesgue measure. AMS Subject Classification: Primary 28D03; Seco...

In this article we present a generalization of the concept of function of bounded variation, in the sense of Riesz, for functions defined on a rectangle in R, which take values in a Banach space. As applications, we obtain generalizations of some results due to Chistyakov and a counterpart of the classical Riesz’s Lemma.

We design an adaptive wavelet scheme for solving first order system least squares formulations of second order elliptic PDEs that converge with the best possible rate in linear complexity. A wavelet Riesz basis is constructed for the space ~ H0,ΓN (div; Ω) on general polygons. The theoretical findings are illustrated by numerical experiments.

Fractional-order dynamics in physics, particularly when applied to diffusion, leads to an extension of the concept of Brownian motion through a generalization of the Gaussian probability function to what is termed anomalous diffusion. As magnetic resonance imaging is applied with increasing temporal and spatial resolution, the spin dynamics is being examined more closely; such examinations exte...