Preprint Series 1999

1999:14
V. Buslaev
The paper is dedicated to the generalizations and applications of Poincaré's theorem on recurrence equations with limit constant coefficients. In particular, applications in the theory of continued fractions, mainly to problems related with Van Vleck’s theorem on regular Cfractions with limit constant coefficients are considered. Special attention is ...
[Full Abstract] 
1999:13
R. Howard,
G. Károlyi, and
L. Székely
We study the possibility of the existence of a Katona type proof for the ErdősKoRado theorem for 2 and 3intersecting families of sets. An ErdősKoRado type theorem for 2intersecting integer arithmetic progressions and a model theoretic argument show that such an approach works in the 2intersecting case.
[Full Abstract] 
1999:12
S. Dilworth,
R. Howard, and
J. Roberts
Let X be a normed space. A set $A\subseteq{X}$ is approximately convex if $d(ta+(1t)b,A)\leq1$ for all $a,b\in{A}$ and $t\in[0,1]$. We prove that every ndimensional normed space contains approximately convex sets A with $H(A,\textrm{Co ...
[Full Abstract] 
1999:11
A. Fokas and
L. Sung
A general approach to solving boundary value problems for two dimensional and integrable nonlinear PDEs was announced in [2] and further developed in [3,4]. This method can be applied to linear PDEs with constant coefficients and to integrable nonlinear PDEs. It involves (a) Formulating the given PDE as the ...
[Full Abstract] 
1999:10
R. DeVore and
G. Petrova
Averaging lemmas deduce smoothness of velocity averages, such as
$$\bar{f}(x):=\int _ \Omega{f}(x,v)dv,\ \ \Omega\subset{\mathbb{R}^d},$$
from properties of f. A canonical example is that $\bar{f}$ is in the Sobolev space $W^{\frac{1}{2}}(L _ 2(\mathbb{R}^d ...
[Full Abstract] 
1999:09
A. Cohen,
W. Dahmen,
I. Daubechies, and
R. DeVore
Tree approximation is a new form of nonlinear approximation which appears naturally in some applications such as image processing and adaptive numerical methods. It is somewhat more restrictive than the usual nterm approximation. We show that the restrictions of tree approximation cost little in terms of rates of approximation. We ...
[Full Abstract] 
1999:08
V. Temlyakov
We suggest a three step strategy to find a good basis (dictionary) for nonlinear mterm approximation. The first step consists of solving an optimization problem for a given function class F, when we optimize over a collection D of bases (dictionaries). The second step is devoted to finding a ...
[Full Abstract] 
1999:07
S. BrennerConvergence of the multigrid Vcycle algorithm for second order boundary value problems without full elliptic regularity (file not available)
(Math. Comp. 71 (2002), 507525)
The multigrid Vcycle algorithm using the Richardson relaxation scheme as the smoother is studied in this paper. For secondorder elliptic boundary value problems, the contraction number of the Vcycle algorithm is shown to improve uniformly with the increase of the number of smoothing steps, without ...
[Full Abstract] 
1999:06
G. Kyriazis and
P. Petrushev
We give a new method for construction of unconditional bases for general classes of TriebelLizorkin and Besov spaces. These include the $L _ p$, $H _ p$, potential, and Sobolev spaces. The main feature of our method is that the character of the basis functions can be prescribed in a ...
[Full Abstract] 
1999:05
É. Czabarka,
G. Konjevod,
M. Marathe,
A. Percus, and
D. TorneyAlgorithms for optimizing production DNA sequencing (file not available)
We discuss the problem of optimally "finishing" a partially sequenced, reconstructed DNA segment. At first sight, this appears to be computationally hard. We construct a series of increasingly realistic models for the problem and show that all of these can in fact be solved to optimality in polynomial time, with ...
[Full Abstract] 
1999:04
A. Cohen,
R. DeVore,
G. Kerkyacharian, and
D. Picard
In recent years, various nonlinear methods have been proposed and deeply investigated in the context of nonparametric estimation: shrinkage methods [21], locally adaptive bandwidth selection [16] and wavelet thresholding [7].
One way of comparing the performances of two different method is to fix a class of functions to be estimated ...
[Full Abstract] 
1999:03
V. Temlyakov
Theoretical greedy type algorithms are studied: a Weak Greedy Algorithm, a Weak Orthogonal Greedy Algorithm, and a Weak Relaxed Greedy Algorithm. These algorithms are defined by weaker assumptions than their analogs the Pure Greedy Algorithm, an Orthogonal Greedy Algorithm, and a Relaxed Greedy Algorighm. The weaker assumptions make these new ...
[Full Abstract] 
1999:02
M. Steel and
L. Székely
In this paper we study how to invert random functions under different criteria. The motivation for this study is phylogeny reconstruction, since the evolution of biomolecular sequences may be considered as a random function from the set of possible phylogenetic trees to the set of collections of biomolecular sequences of ...
[Full Abstract] 
1999:01
R. Getsadze
A complement to A.M. Olevskii’s fundamental inequality on logarithmic growth of Lebesgue functions of an arbitrary uniformly bounded orthonormal system on a set of positive measure is made. Namely, the index where the Lebesgue functions have growth slightly weaker than logarithm can be chosen independent of the variable ...
[Full Abstract]