Preprint Series 2012

2012:09
K. Kazarian and
V. Temlyakov
We study Hilbert spaces $\mathfrak{L}^{2}(E,G)$, where $E\subset \mathbb{R}^{d}$ is a measurable set, $E>0$ and for almost every $t\in E$ the matrix $G(t)$ (see (3)) is a Hermitian positivedefinite matrix. We find necessary and sufficient conditions for which the projection operators ...
[Full Abstract] 
2012:08
G. Kerkyacharian and
P. Petrushev
Classical and nonclassical Besov and TriebelLizorkin spaces with complete range of indices are developed in the general setting of Dirichlet space with a doubling measure and local scaleinvariant Poincaré inequality. This leads to Heat kernel with small time Gaussian bounds and HÃ¶lder continuity, which play a central role in this ...
[Full Abstract] 
2012:07
P. Binev,
A. Cohen,
W. Dahmen, and
R. DeVore
Algorithms for binary classification based on adaptive partitioning are formulated and analyzed for both their risk performance and their friendliness to numerical implementation. The algorithms can be viewed as generating a set approximation to the Bayes set and thus fall into the general category of set estimators. A general ...
[Full Abstract] 
2012:06
T. Coulhon,
G. Kerkyacharian, and
P. Petrushev
Wavelet bases and frames consisting of band limited functions of nearly exponential localization on $\mathbb{R}^d$ are a powerful tool in harmonic analysis by making various spaces of functions and distributions more accessible for study and utilization, and providing sparse representation of natural function spaces (e.g. Besov spaces ...
[Full Abstract] 
2012:05
V. Temlyakov
This paper is a follow up to the previous author's paper on convex optimization. In that paper we began the process of adjusting greedytype algorithms from nonlinear approximation for finding sparse solutions of convex optimization problems. We modified there three the most popular in nonlinear approximation in Banach spaces ...
[Full Abstract] 
2012:04
J. Nelson and
V. Temlyakov
We study rate of convergence of expansions of elements in a Hilbert space $H$ into series with regard to a given dictionary ${\cal D}$.
[Full Abstract]
The primary goal of this paper is to study representations of an element $f\in H$ by a series
$$f\sim \displaystyle\sum\limits _ {j ... 
2012:03
V. Temlyakov
We study sparse approximate solutions to convex optimization problems. It is known that in many engineering applications researchers are interested in an approximate solution of an optimization problem as a linear combination of elements from a given system of elements. There is an increasing interest in building such sparse approximate ...
[Full Abstract] 
2012:02
S. Dilworth,
M. SotoBajo, and
V. Temlyakov
We study Lebesguetype inequalities for greedy approximation with respect to quasigreedy bases. We mostly concentrate on this study in the $L _ p$ spaces. The novelty of the paper is in obtaining better Lebesguetype inequalities under extra assumptions on a quasigreedy basis than known Lebesguetype inequalities for quasigreedy bases. We ...
[Full Abstract] 
2012:01
D. Savu and
V. Temlyakov
We study sparse representations and sparse approximations with respect to incoherent dictionaries. We address the problem of designing and analyzing greedy methods of approximation. A key question in this regard is: How to measure efficiency of a specific algorithm? Answering this question we prove the Lebesguetype inequalities for algorithms under ...
[Full Abstract]