Greedy-Type Approximation in Banach Spaces and Applications

We continue to study the efficiency of approximation and convergence of greedy-type algorithms in uniformly smooth Banach spaces. Two greedy-type approximation methods, the Weak Chebyshev Greedy Algorithm (WCGA) and the Weak Relaxed Greedy Algorithm (WRGA), have been introduced and studied in [24]. These methods (WCGA and WRGA) are very general approximation methods that work well in an arbitrary uniformly smooth Banach space $X$ for any dictionary ${\Cal D}$. It turns out that these general approximation methods are also very good for specific dictionaries. It has been observed in [7] that the WCGA and WRGA provide constructive methods in $m$-term trigonometric approximation in $L_p$, $p\in[2,\infty)$, which realize an optimal rate of $m$-term approximation for different function classes. In [25] the WCGA and WRGA have been used in constructing deterministic cubature formulas for a wide variety of function classes with error estimates similar to those for the Monte Carlo Method. The WCGA and WRGA can be considered as a constructive deterministic alternative to (or substitute for) some powerful probabilistic methods. This observation encourages us to continue a thorough study of the WCGA and WRGA. In this paper we study modifications of the WCGA and WRGA that are motivated by numerical applications. In these modifications we are able to perform steps of the WCGA (or WRGA) approximately with some controlled errors. We prove that the modified versions of the {\it WCGA and WRGA perform as well as the WCGA and WRGA}. We give two applications of greedy-type algorithms. First, we use them to provide a constructive proof of optimal estimates for best $m$-term trigonometric approximation in the uniform norm. Second, we use them to construct deterministic sets of points $\{\xi^1,\ldots,\xi^m\} \subset [0,1]^d$ with the $L_p$ discrepancy less than $Cp^{1/2}m^{-1/2}$, $C$ is an effective absolute constant.