Preprint Series 2001

2001:26
R. DeVore,
G. Petrova, and
V. Temlyakov
We study the approximation of a function class F in $L _ p$ by choosing first a basis B and then using nterm approximation with the elements of B. Into the competition for best bases we enter all greedy (i.e. democratic and unconditional [20]) bases for $L _ ...
[Full Abstract] 
2001:25
I. Daubechies,
R. DeVore,
C. Güntürk, and
V. Vaishampayan
We analyze mathematically the effect of quantization error in the circuit implementation of Analog to Digital (A/D) converters such as Pulse Code Modulation (PCM) and SigmaDelta Modulation (∑∆). We show that ∑∆ modulation, which is based on oversampling the signal, has a self correction for quantization error that is not inherited ...
[Full Abstract] 
2001:24
I. Daubechies and
R. DeVore
Digital signal processing has revolutionized the storage and transmission of audio signals, images and video, in consumer electronics as well as in more scientific settings (such as medical imaging). The main advantage of digital signal processing is its robustness: although all the operations have to be implemented with necessarily not ...
[Full Abstract] 
2001:23
S. Dilworth,
N. Kalton,
D. Kutzarova, and
V. Temlyakov
Some new conditions that arise naturally in the study of the Thresholding Greedy Algorithm are introduced for bases of Banach spaces. We relate these conditions to best nterm approximation and we study their duality theory. In particular, we obtain a complete duality theory for greedy bases.
[Full Abstract] 
2001:22
J. Griggs
We determine the minimum volume (sum of cardinalities) of an intersecting family of subsets of an nset, given the size of the family, by solving a simple linear program. From this we obtain a lower bound on the average size of the sets in an intersecting family. This answers ...
[Full Abstract] 
2001:21
S. BrennerSmoothers, mesh dependent norms, interpolation and multigrid (file not available)
(Appl. Numer. Math. 43 (2002), 4556.)
New estimates for the nodal interpolation operator for the P
[Full Abstract]finite element are established with respect to mesh dependent norms that are defined in terms of the smoothers in multigrid algorithms. These estimates are useful in the additive approach to the convergence of Vcycle ... 
2001:20
H. Wang
In this paper, we analyze the modified method of characteristics (MMOC) and an improved version of the MMOC, named the modified method of characteristics with adjusted advection (MMOCAA), for multidimensional advectionreaction transport equations in a uniform manner. We derive an optimalorder error estimate for these schemes. Numerical results are presented ...
[Full Abstract] 
2001:19
J. Liu and
H. Wang
We combine an EulerianLagrangian approach and multiresolution analysis to develop unconditionally stable, explicit, multilevel methods for multidimensional linear hyperbolic equations. The derived schemes generate accurate numerical solutions even if large time steps are used. Furthermore, these schemes have the capability of carrying out adaptive compression without introducing mass balance error ...
[Full Abstract] 
2001:18
M. Fischermann,
A. Hoffmann,
D. Rautenbach,
L. Székely, and
L. Volkmann
The Wiener index of a graph is the sum of all pairwise distances of vertices of the graph. In this paper we characterize the trees which minimize the Winer index among all trees of given order and maximum degree and the trees which maximize the Wiener index among all trees ...
[Full Abstract] 
2001:17
E. Matouskova
Let f be a biLipschitz mapping of the Euclidean ball $B _ {R^n}$ into $\ell _ 2$ with both Lipschitz constants close to one. We investigate the shape of $f(B _ {R^n}$). We give examples of such a mapping f, which has the Lipschitz constants arbitrarily close ...
[Full Abstract] 
2001:16
R. Anstee,
R. Ferguson, and
J. Griggs
Consider the permutation $\pi=(\pi _ 1,\ldots,\pi _ n)$ of $1,2,\ldots,n$ as being placed on a circle with indices taken modulo n. For given $k<n$ there are n sums of k consecutive entries, and their average is $\frac{k(n+1)}{2}$. We say ...
[Full Abstract] 
2001:15
S. BrennerConvergence of nonconforming Vcycle and Fcycle multigrid algorithms for second order elliptic boundary value problems (file not available)
(Mathematics of Computation 73 (2004), 10411066, electronically posted August 19, 2003, PII: S00255718(03)015783)
The convergence of Vcycle and Fcycle multigrid algorithms with a sufficiently large number of smoothing steps is established for nonconforming finite element methods for second order elliptic boundary value problems.
[Full Abstract] 
2001:14
S. Dilworth,
D. Kutzarova, and
V. Temlyakov
We consider some theoretical greedy algorithms for approximation in Banach spaces with respect to a general dictionary. We prove convergence of the algorithms for Banach spaces which satisfy certain smoothness assumptions. We compare the algorithms and their rates of convergence when the Banach space is $L _ p(T^d ...
[Full Abstract] 
2001:13
B. Karaivanov and
P. Petrushev
We study nonlinear nterm approximation in $L _ p(R^2)$ ($0<p<\infty$) from Courant elements or (discontinuous) piecewise polynomials generated by multilevel nested triangulations of $R^2$ which allow arbitrarily sharp angles. To characterize the rate of approximation we introduce and develop three families of smoothness spaces ...
[Full Abstract] 
2001:12
P. Petrushev
We study nonlinear approximation in $L _ p(R^d)\ (0<p<\infty,>1)$ from (a) nterm rational functions, and (b) piecewise polynomials generated by different anisotropic dyadic partitions of $R^d$. To characterize the rates of each such piecewise polynomial approximation we introduce a family of smoothness spaces ...
[Full Abstract] 
2001:11
M. AlLawatia and
H. Wang
We develop an EulerianLagrangian substructuring domain decomposition method for the solution of unsteadystate advectiondiffusion transport equations. This method reduces to an EulerianLagrangian scheme within each subdomain and to a type of DirichletNeumann algorithm at subdomain interfaces. The method generates accurate and stable solutions that are free of artifacts even if ...
[Full Abstract] 
2001:10
M. Steel and
L. Székely
In this paper we study inverting random functions under the maximum likelihood estimation (MLE) criterion. In particular, we consider how many independent evaluations of the random function at a particular element of the domain are needed for reliable reconstruction of that element. We provide explicit upper and lower bounds for ...
[Full Abstract] 
2001:09
V. Temlyakov
Our main interest in this paper is nonlinear approximation. The basic idea behind nonlinear approximation is that the elements used in the approximation do not come from a fixed linear space but are allowed to depend on the function being approximated. While the scope of this paper is mostly theoretical ...
[Full Abstract] 
2001:08
O. Trifonov
An approach of SwinnertonDyer is extended to obtain new upper bounds for the number of lattice points close to a smooth curve. One consequence of these bounds is a new asymptotic result for the distribution of squarefull numbers in short intervals.
[Full Abstract] 
2001:07
V. Temlyakov
We discuss one lower estimate for the rate of convergence of Pure Greedy Algorithm with regard to a general dictionary and another lower estimate for the rate of convergence of Weak Greedy Algorithm with a special weakness sequence $\tau =\{t\}$, $0<t<1$, with regard to a general dictionary. The ...
[Full Abstract] 
2001:06
M. Filaseta and
D. Meade
For $w(x)\in{C}[x]$ with $w(x)\not\equiv0$, define the reciprocal of $w(x)$ to be the polynomial
$$\tilde{w}(x)=x^{\deg{w}}w(\frac{1}{x}).$$
We refer to $w(x)$ as being reciprocal if $w(x)=\pm\tilde{w}(x)$ and as being nonreciprocal ...
[Full Abstract] 
2001:05
M. Skopina
Wavelettype systems providing a uniformly convergent expansion for any continuous function on the sphere are found. This construction is transferred to the disk due to some special connections between polynomial bases on the sphere and on the disk.
[Full Abstract] 
2001:04
S. Dilworth,
R. Howard, and
J. Roberts
Let $\Delta _ m=\{(t _ 0,\ldots,t _ m)\in{R^{n+1}}:t _ i\geq0,\sum^m _ {i=0}t _ i=1\}$ be the standard mdimensional simplex. Let $\emptyset\neq{S}\subset\bigcup^\infty _ {m=1}\Delta _ m$, then a function ...
[Full Abstract] 
2001:03
S. BrennerA new look at FETI (file not available)
(Proceedings of the Thirteenth International Conference on Domain Decomposition Methods, N. Debit, M. Garbey, R. Hoppe, J. Périaux, D. Keyes and Y. Kuznetsov, ed., DDM.org, 2001, pp. 4151)
A coordinatefree formulation of the Finite Element Tearing and Interconnecting (FETI) method and a new 3D FETI preconditioner are presented in ...
[Full Abstract] 
2001:02
M. Nielsen and
D. Zhou
We study the mean size of wavelet packets in L_{p}. An exact formula for the mean size is given in terms of the pnorm joint spectral radius. This will be a corollary of an asymptotic formula for the L_{p}norms on the subdivision trees. Then the stability ...
[Full Abstract] 
2001:01
S. Brenner and
Q. HeLower bounds for threedimensional nonoverlapping domain decomposition algorithms (file not available)
(Numerische Mathematik 93 (2003), 445470)
Lower bounds for the condition numbers of the preconditioned systems are obtained for the wire basket preconditioner and the NeumannNeumann preconditioner in three dimensions. They show that the known upper bounds are sharp.
[Full Abstract]