Preprint Series 2000

2000:30
V. Konovalov and
D. Leviatan
Let I be a finite interval, $r\in{\mathbb{N}}$ and $\rho(t)=\textrm{dist}\{t,\delta{I}\}$, $t\in{I}$. Denote by $\Delta^s+W^r _ {p,\alpha},0\leq{\alpha}<\infty$, the class of functions x on I with the seminorm $\x^{(r)}\rho^\alpha\L _ ...
[Full Abstract] 
2000:29
S. Dekel and
D. LeviatanWavelet decompositions of nonrefinable shift invariant spaces (file not available)
The motivation for this work is a recently constructed family of generators of shiftinvariant spaces with certain optimal approximation properties, but which are not refinable in the classical sense. We try to see whether, once the classical refinability requirement is removed, it is still possible to construct meaningful wavelet decompositions ...
[Full Abstract] 
2000:28
A. Schinzel
K. Oskolkov has called my attention to the following theorem used in the theory of polynomial approximation ([2], forumula(16)): for every sequence of $d+1$ pairwise linearly independent vectors $[\alpha _ {\mu1},\alpha _ {\mu2}]\in{C}^2(1\leq\mu\leq{d}+1)$ and every polynomial $F\in ...
[Full Abstract] 
2000:27
P. Chrusciel,
J. Fu,
J. Galloway, and
R. Howard
We study fine differentiability properties of horizons. We show that the set of end points of generators of a ndimensional horizon H (which is included in a $(n+1)$dimensional spacetime M) has vanishing ndimensional Hausdorff measure. This is proved by showing that the set of end points ...
[Full Abstract] 
2000:26
S. Dilworth,
R. Howard, and
J. Roberts
Let $n\geq1$ and $B\geq2$. A realvalued function f defined on the nsimplex $\Delta _ n$ is approximately convex with respect to $\Delta _ {B1}$ if
$$f\left(\sum^B _ {i=1}t _ ix _ i\right)\leq\sum^B _ {i=1}t _ if ...
[Full Abstract] 
2000:25
A. Abrams,
J. Cantarella,
J. Fu,
M. Ghomi, and
R. Howard
We show that circles uniquely minimize most of O'Hara's knot energies, including those conjectured by Freedman, He, and Wang. The proof is based on a theorem of Lükő on average chord lengths of closed curves. We also prove this result for a broader class of energy functionals and ...
[Full Abstract] 
2000:24
M. Skopina
In the weighted space $L _ {2,w}(B)$, where $w(x)=\pi^{1}(1x^2)^{\frac{1}{2}}$, B is the unit disk in $R^2$, a complete orthogonal system of ridge polynomials is constructed. Ridge directions of the polynomials are condensed. Sufficient conditions for uniform convergence and ...
[Full Abstract] 
2000:23
R. Gribonval and
M. Nielsen
We study the approximation classes $A _ {\alpha,s}$ associated with nonlinear mterm approximation by elements from a quasinormed Schauder basis in a separable Banach space. We show that there always is a twosided embedding
$K _ {\tau _ l,s}\hookrightarrow{A} _ {\alpha,s}\hookrightarrow{K} _ ...
[Full Abstract] 
2000:22
F. Belgacem and
S. BrennerSome nonstandard finite element estimates with applications to 3D Poisson and Signorini problems (file not available)
(Electronic Transactions on Numerical Analysis 12 (2001), 134148)
In this paper we establish several nonstandard finite element estimates involving fractional order Sobolev spaces, with applications to bubble stabilized mixed methods for the threedimensional Poisson and Signorini problems.
[Full Abstract] 
2000:21
V. Temlyakov
This paper completes the investigation of necessary and sufficient conditions on the "weakness" sequence $\tau:=\{t _ k\}^\infty _ {k=1}$ for convergence of Weak Greedy Algorithm for all dictionaries D and each function (vector) f in Hilbert space H. This paper is a follow up to the papers ...
[Full Abstract] 
2000:20
R. Gribonval
In a mixture of two highdimensional Gaussian classes, the class can be identified with active testing, i.e. by a (sequential) adaptive selection of features from a redundant dictionary. Using the mutual entropy criterion, we provide an analytic characterization of this procedure in two situations. When the classes have the ...
[Full Abstract] 
2000:19
L. Székely
This paper surveys how the concept of crossing number, which used to be familiar only to a limited group of specialists, emerges as a significant graph parameter. This paper has dual purposes: first, it reviews foundational, historical, and philosophical issues of crossing numbers, second, it shows a new lower bound ...
[Full Abstract] 
2000:18
L. Székely
This is a concise review that attempts to show the vast influence of the work of Paul Erdős in a narrow area, the combinatorics of unit distances in geometry. This review tries to follow the history of the problems and cover the latest and strongest results, but cannot be complete ...
[Full Abstract] 
2000:17
A. Petukhov
The paper deals with finding criteria for the Hausdorff convergence of sequences of convolution operators on quasiBanach spaces of periodic realvalued distributions (generalized functions). In particular, the criteria for convergence on the Hardy classes, on the class of regular Borel measures, and on the class of pseudomeasures are found. These ...
[Full Abstract] 
2000:16
R. Gribonval and
M. Nielsen
We present a generalization of V. Temlyakov's weak greedy algorithm, and give a sufficient condition for norm convergence of the algorithm for an arbitrary dictionary in a Hilbert space. We provide two counterexamples to show that the condition cannot be relaxed for general dictionaries. For a class of dictionaries ...
[Full Abstract] 
2000:15
A. Petukhov
We study tight wavelet frames associated with symmetric compactly supported refinable functions, which are obtained with the unitary extension principle. We give a criterion for existence of two symmetric or antisymmetric compactly supported framelets.
All refinable masks of length up to 6, satisfying this criterion, are found.
[Full Abstract] 
2000:14
G. Nikolov
We prove here that if an algebraic polynomial f of degree at most n has smaller absolute values than T_{n} (the nth Chebyshev polynomial of the first kind) at arbitrary n + 1 points in [1,1], which interlace with the zeros of T_{n}, then the uniform norm ...
[Full Abstract] 
2000:13
E. Livshitz and
V. Temlyakov
We study the convergence, in a Hilbert space, of a Weak Greedy Algorithm (WGA) which is a modification of a Pure Greedy Algorithm (PGA). At the mth step of a WGA, we choose an approximating element from a given dictionary D satisfying the relation $\langle{f}^\tau _ {m1 ...
[Full Abstract] 
2000:12
R. Gribonval
In a separable Hilbert space H, greedy algorithms iteratively define mterm approximants to a given vector from a complete redundant dictionary D. With very large dictionaries, the pure greedy algorithm cannot be implemented and must be replaced with a weak greedy algorithm. A conjecture about the convergence of very ...
[Full Abstract] 
2000:11
M. Nielsen
We introduce wavelet packets in the setting of a multiresolution analysis of $L^2(R^d)$ generated by an arbitrary dilation matrix A satisfying $\det{A}=2$, and note that all the analysis and algorithms of wavelet packets in the standard one dimensional case can be generalized to this multidimensional ...
[Full Abstract] 
2000:10
V. Temlyakov
We study efficiency of approximation and convergence of two greedy type algorithms in uniformly smooth Banach spaces. The Weak Chebyshev Greedy Algorithm (WCGA) is defined for an arbitrary dictionary D and provides nonlinear mterm approximation with regard to D. This algorithm is defined inductively with the mth step ...
[Full Abstract] 
2000:09
V. Maiorov,
K. Oskolkov, and
V. Temlyakov
Gridge approximation compiles greedy algorithms and ridge approximation. It is a class of algorithmic constructions of ridge functions  finite linear combinations of planar waves. The goal is to approximate a given target which is a multivariate function. On each step, a new planar wave is added to the preceding linear ...
[Full Abstract] 
2000:08
R. Howard
On any open subset U of the Euclidean space $R^n$ there is complete torsionfree connection whose geodesics are reparameterizations of the intersections of the straight lines of $R^n$ with U. For any positive integer m there is a complete projectively flat torsion free connection on the two dimensional ...
[Full Abstract] 
2000:07
E. Cornea,
R. Howard, and
P. Martinsson
A detailed study of solutions to the first order partial differential equation $H(x,y,z _ x,z _ y)=0$, with special emphasis on the eikonal equation $z^2 _ x+z^2 _ y=h(x,y)$, is made near points where the equation becomes singular in ...
[Full Abstract] 
2000:06
S. BrennerAn additive Schwarz preconditioner for the FETI method (file not available)
(Numerische Mathematik 94 (2003), 131)
A new additive Schwarz preconditioner for the Finite Element Tearing and Interconnecting (FETI) method is analyzed in this paper. This preconditioner has the unique feature that the coefficient matrix of its "coarse grid" problem is mesh independent. For a model second order heterogeneous elliptic boundary ...
[Full Abstract] 
2000:05
S. Brenner and
L. SungDiscrete Sobolev and Poincaré inequalities via Fourier series (file not available)
(EastWest Journal of Numerical Mathematics 8 (2000), 8392)
Discrete Sobolev and Poincaré inequalities for finite element functions are derived and proved to be sharp via Fourier series.
[Full Abstract] 
2000:04
T. Elbouayachi
This article is a detailed study of a class of indefinitely oscillating functions in $H^s(\mathbb{R})$. It's a class of functions of Sobolev space $H^s(\mathbb{R})$ which have for all m integer one primitive of the order m in the same space.
[Full Abstract] 
2000:03
A. Petukhov
We study tight wavelet frames and biframes, associated with given scaling functions, which are obtained with the unitary and mixed extension principles. All possible solutions of the corresponding matrix equations are found. It is proved that the problem of the extension may be always solved with two framelets. In particular ...
[Full Abstract] 
2000:02
R. Gribonval
We introduce a modified Matching Pursuit algorithm, called Fast Ridge Pursuit, to approximate Ndimensional signals with M Gaussian chirps at a computational cost O(MN) instead of the expected O(MN^{2} log N). At each iteration of the pursuit, the best Gabor atom is first selected, then its scale ...
[Full Abstract] 
2000:01
P. Chrusciel,
E. Delay,
J. Galloway, and
R. Howard
We prove that the area of sections of future event horizons in space times satisfying the null energy condition is nondecreasing toward the future under the following circumstances: 1) the horizon is future geodesically complete; 2) the horizon is a black hole event horizon in a globally hyperbolic spacetime and ...
[Full Abstract]