This paper considers recovery of signals that are sparse or approximately sparse in terms of a general frame from undersampled data corrupted with additive noise. We show that the properly constrained [Formula: see text]-analysis, called general-dual-based analysis Dantzig selector, stably recovers a signal which is nearly sparse in terms of a general dual frame provided that the measurement matrix satisfies a restricted isometry property adapted to the general frame. As a special case, we consider the Gaussian noise.

In this paper, we present a monotone domain decomposition iterative technique for boundary value problems of a nonlinear fractional differential equation. We construct two monotone domain decomposition sequences of upper and lower solutions which converge uniformly to the actual solution of the problem. The accuracy and efficiency of our new approach are demonstrated through two examples.

We present a method constructing a function which is the best approximation for given data and satisfiesthe given self-similar condition. For this, we construct a space F of local self-similar fractal functions and show its properties. Next we present a computational scheme constructing the best fractal approximation in this space and estimate an error of the constructed fractal approximation. Our best fractal approximation is a fixed point of some fractal interpolation function.