Preprint Series 2006

2006:16
F. Narcowich,
P. Petrushev, and
J. Ward
A discrete system of almost exponentially localized elements (needlets) on the $n$dimensional unit sphere $\mathbb{S}^n$ is constructed. It is shown that the needlet system can be used for decomposition of Besov and TriebelLizorkin spaces on the sphere. As an application of Besov spaces on $\mathbb{S}^n ...
[Full Abstract] 
2006:15
É. Czabarka and
J. Spouge
An urn contains white and black balls numbered arbitrarily from 1 to $d$ . The tendency of white balls to have lower numbers than black balls can be measured by a “$ROC _ n$ score”. Now, let each ball have not one but two numbers on it, in red and green ...
[Full Abstract] 
2006:14
L. Lu
Let $f(r)=\min _ H\sum _ {f\in{E(H)}}\frac{1}{2^{F}}$, where H ranges over all 3chromatic hypergraphs with minimum edge cardinality r. ErdősLovász (1975) conjectured $f(r)\to\infty$ as $r\to\infty$. This conjecture was proved by Beck in 1978. Here we show ...
[Full Abstract] 
2006:13
L. Lu and
L. Székely
The Lovász Local Lemma is known to have an extension for cases where independence is missing but negative dependencies are under control [1]. We show that this is often the case for random injections, and we provide easytocheck conditions for the nontrivial task of verifying a negative dependency graph for ...
[Full Abstract] 
2006:12
M. Steel and
L. Székely
This paper continues our earlier investigations into the inversion of random functions in a general (abstract) setting. In Section 2 we investigate a concept of invertibility and the invertibility of the composition of random functions. In Section 3 we resolve some questions concerning the number of samples required to ensure ...
[Full Abstract] 
2006:11
G. Kyriazis,
P. Petrushev, and
Y. Xu
The LittlewoodPaley theory is extended to weighted spaces of distributions on $[1,1]$ with Jacobi weights $w(t)=(1t)^\alpha(1+t)^\beta$. Almost exponentially localized polynomial elements (needlets) $\{\varphi _ \xi\},\{\psi _ \xi\}$ are constructed and, in complete analogy with the classical case on $R^n$, it is ...
[Full Abstract] 
2006:10
S. Zheng
Let H be a Schrödinger operator on $R^n$. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and TriebelLizorkin spaces associated with H are well defined. We further give a LittlewoodPaley characterization of $L _ p$ spaces in terms of ...
[Full Abstract] 
2006:09
G. Kerkyacharian,
P. Petrushev,
D. Picard, and
T. Willer
We provide a new algorithm for the treatment of inverse problems which combines the traditional SVD inversion with an appropriate thresholding technique in a well chosen new basis. Our goal is to devise an inversion procedure which has the advantages of localization and multiscale analysis of wavelet representations without losing ...
[Full Abstract] 
2006:08
P. Petrushev and
Y. Xu
Almost exponentially localized polynomial kernels are constructed on the unit ball $B^d$ in $R^d$ with weights $W _ \mu(x)=(1x^2)^{\mu\frac{1}{2}},$$\mu\geq0$, by smoothing out the coefficients of the corresponding orthogonal projectors. These kernels are utilized to the design of cubature ...
[Full Abstract] 
2006:07
V. Temlyakov
We study greedy algorithms in a Banach space from the point of view of convergence and rate of convergence. We concentrate on studying algorithms that provide expansions into a series. We call such expansions greedy expansions. It was pointed out in our previous paper that there is a great flexibility ...
[Full Abstract] 
2006:06
W. Dahmen,
S. Dekel, and
P. Petrushev
This paper is concerned with the construction and analysis of multilevel Schwarz preconditioners for partition of unity methods applied to elliptic problems. We show under which conditions on a given multilevel partition of unity hierarchy (MPUM) one even obtains uniformly bounded condition numbers and how to realize such requirements. The ...
[Full Abstract] 
2006:05
G. Olafsson and
S. Zheng
We address the function space theory associated with the Schrödinger operator $H=\frac{d^2}{dx^2}+V$. The discussion is featured with potential $V(x)=$$n(n+1)\textrm{sech}^2x$, which is called in quantum physics PöschlTeller potential. Using a dyadic system, we introduce TriebelLizorkin spaces and Besov ...
[Full Abstract] 
2006:04
É. Czabarka,
O. Sykora,
L. Székely, and
I. Vrto
The biplanar crossing number $cr _ 2(G)$ of a graph G is $\min\{cr(G _ 1)+cr(G _ 2)\}$, where cr is the planar crossing number and $G _ 1\cup{G _ 2}=G$. We show that $cr _ 2(G)\leq(\frac{3}{8})cr ...
[Full Abstract] 
2006:03
V. Temlyakov
We study greedy algorithms in a Banach space from the point of view of convergence and rate of convergence. There are two wellstudied approximation methods: the Weak Chebyshev Greedy Algorithm (WCGA) and the Weak Relaxed Greedy Algorithm (WRGA). The WRGA is simpler than the WCGA in the sense of computational ...
[Full Abstract] 
2006:02
L. Ju
As the methodology of centroidal Voronoi tessellation (CVT) is receiving more and more attention in the mesh generation community, a clear characterization of the influence of geometric constraints on the CVTbased meshing is becoming increasingly important. In this paper, we first give a precise definition of the geometrically conforming centroidal ...
[Full Abstract] 
2006:01
D. Donoho,
M. Elad, and
V. Temlyakov
We study the efficiency of greedy algorithms with regard to redundant dictionaries in Hilbert spaces. We obtain upper estimates for the errors of the Pure Greedy Algorithm and the Orthogonal Greedy Algorithm in terms of the best mterm approximations. We call such estimates the Lebesgue type inequalities. We prove ...
[Full Abstract]