共查询到20条相似文献,搜索用时 35 毫秒
1.
In this paper we study the worst-case error (of numerical integration) on the unit sphere
for all functions in the unit ball of the Sobolev space
where
More precisely, we consider infinite sequences
of m(n)-point numerical integration rules
where: (i)
is exact for all spherical polynomials of degree
and (ii)
has positive weights or, alternatively to (ii), the sequence
satisfies a certain local regularity property. Then we show that the worst-case error (of numerical integration)
in
has the upper bound
where the constant c depends on s and d (and possibly the sequence
This extends the recent results for the sphere
by K. Hesse and I.H. Sloan to spheres
of arbitrary dimension
by using an alternative representation of the worst-case error. If the sequence
of numerical integration rules satisfies
an order-optimal rate of convergence is achieved. 相似文献
2.
C. Carton-Lebrun 《Journal of Fourier Analysis and Applications》1995,2(1):49-64
For
define
where
Pointwise estimates and weighted inequalities describing the local Lipschitz continuity
of
are established. Sufficient conditions are found
for the boundedness of
from
into
and a spherical restriction property is proved. A study of the moment subspaces of
is next developed in the one-variable case, for
locally integrable,
a.e. It includes a decomposition theorem and a complete classification of all possible sequences of moment subspaces in
Characterizations are also given for each class. Applications related to the approximation and decomposition of
are discussed. 相似文献
3.
4.
A triangulation of a set S of points in the plane is a subdivision of the convex hull of S into triangles whose vertices are
points of S. Given a set S of n points in
each moving independently, we wish to maintain a triangulation of S. The triangulation needs to be updated periodically as
the points in S move, so the goal is to maintain a triangulation with a small number of topological events, each being the
insertion or deletion of an edge. We propose a kinetic data structure (KDS) that processes
topological events with high probability if the trajectories of input points are algebraic curves of fixed degree. Each topological
event can be processed in
time. This is the first known KDS for maintaining a triangulation that processes a near-quadratic number of topological events,
and almost matches the
lower bound [1]. The number of topological events can be reduced to
if only k of the points are moving. 相似文献
5.
A compact set
is staircase connected if every two points
can be connected by a polygonal path with sides parallel to the coordinate axes, which is both x-monotone and y-monotone.
denotes the smallest number of edges of such a path.
is an integer-valued metric on S. We investigate this metric and introduce stars and kernels. Our main result is that the
r-th kernel is nonempty, compact and staircase connected provided
. 相似文献
6.
Nonlinear Approximation by Trigonometric Sums 总被引:7,自引:0,他引:7
We investigate the
-error of approximation to a function
by a linear combination
of
exponentials
on
where the frequencies
are allowed to depend on
We bound this error in terms of the smoothness and other properties of
and show that our bounds are best possible in the sense of approximation of certain classes of functions. 相似文献
7.
Let
denote the linear space over
spanned by
. Define the (real) inner product
, where V satisfies: (i) V is real analytic on
; (ii)
; and (iii)
. Orthogonalisation of the (ordered) base
with respect to
yields the even degree and odd degree orthonormal Laurent polynomials
, and
. Define the even degree and odd degree monic orthogonal Laurent polynomials:
and
. Asymptotics in the double-scaling limit
such that
of
(in the entire complex plane),
, and
(in the entire complex plane) are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a
matrix Riemann-Hilbert problem on
, and then extracting the large-n behaviour by applying the non-linear steepest-descent method introduced in [1] and further
developed in [2],[3]. 相似文献
8.
Given a collection S of subsets of some set
and
the set cover problem is to find the smallest subcollection
that covers
that is,
where
denotes
We assume of course that S covers
While the general problem is NP-hard to solve, even approximately, here we consider some geometric special cases, where usually
Combining previously known techniques [4], [5], we show that polynomial-time approximation algorithms with provable performance
exist, under a certain general condition: that for a random subset
and nondecreasing function f(·), there is a decomposition of the complement
into an expected at most f(|R|) regions, each region of a particular simple form. Under this condition, a cover of size O(f(|C|))
can be found in polynomial time. Using this result, and combinatorial geometry results implying bounding functions f(c) that
are nearly linear, we obtain o(log c) approximation algorithms for covering by fat triangles, by pseudo-disks, by a family
of fat objects, and others. Similarly, constant-factor approximations follow for similar-sized fat triangles and fat objects,
and for fat wedges. With more work, we obtain constant-factor approximation algorithms for covering by unit cubes in
and for guarding an x-monotone polygonal chain. 相似文献
9.
We give conditions on radial nonnegative weights $W_1We give conditions on radial nonnegative weights
and
on
, for which the a priori inequality
holds with constant independent of
. Here
is the Laplace-Beltrami operator on the sphere
. Due to the relation between
and the tangential component of the gradient,
, we obtain some "Morawetz-type" estimates for
on
. As a consequence we establish some new estimates for the free Schr?dinger propagator
, which may be viewed as certain refinements of the
-(super)smoothness estimates of Kato and Yajima. These results, in turn, lead to the well-posedness of the initial value problem
for certain time dependent first order spherical perturbations of the
dimensional Schr?dinger equation. 相似文献
10.
Anders Nedergaard Jensen 《Discrete and Computational Geometry》2007,37(3):443-453
The Grobner fan of an ideal
, defined by Mora and Robbiano, is a complex of polyhedral cones in
. The maximal cones of the fan are in bijection with the distinct monomial initial ideals of I as the term order varies. If
I is homogeneous the Grobner fan is complete and is the normal fan of the state polytope of I. In general the Grobner fan
is not complete and therefore not the normal fan of a polytope. We may ask if the restricted Grobner fan, a subdivision of
, is regular, i.e. the normal fan of a polyhedron. The main result of this paper is an example of an ideal in n = 4 variables
whose restricted Grobner fan is not regular. 相似文献
11.
Daniel A. Klain 《Discrete and Computational Geometry》2006,36(3):457-477
Hyperbolic area is characterized as the unique continuous isometry-invariant simple valuation on convex polygons in
We then show that continuous isometry-invariant simple valuations on polytopes in
for
are determined uniquely by their values at ideal simplices. The proofs exploit a connection between valuation theory in
hyperbolic space and an analogous theory on the Euclidean sphere. These results lead to characterizations of continuous isometry-invariant
valuations on convex polytopes and convex bodies in the hyperbolic plane
a partial characterization in
and a mechanism for deriving many fundamental theorems of hyperbolic integral geometry, including kinematic formulas,
containment theorems, and isoperimetric and Bonnesen-type inequalities. 相似文献
12.
James H. Schmerl 《Discrete and Computational Geometry》2007,38(1):155-167
Suppose n ≥ 2 and
are such that X is infinite and Y is the set of vertices of an n-simplex. Then there is blue/red coloring of
such that no set similar to X is monochromatically blue and no set similar to Y is monochromatically red. 相似文献
13.
Oswin Aichholzer Jesus Garcia David Orden Pedro Ramos 《Discrete and Computational Geometry》2007,38(1):1-14
We provide a new lower bound on the number of (≤ k)-edges of a set of n points in the plane in general position. We show that
for
the number of (≤ k)-edges is at least
which, for
, improves the previous best lower bound in [12]. As a main consequence, we obtain a new lower bound on the rectilinear crossing
number of the complete graph or, in other words, on the minimum number of convex quadrilaterals determined by n points in
the plane in general position. We show that the crossing number is at least
which improves the previous bound of
in [12] and approaches the best known upper bound
in [4]. The proof is based on a result about the structure of sets attaining the rectilinear crossing number, for which we
show that the convex hull is always a triangle. Further implications include improved results for small values of n. We extend
the range of known values for the rectilinear crossing number, namely by
and
. Moreover, we provide improved upper bounds on the maximum number of halving edges a point set can have. 相似文献
14.
Given a function ψ in
the affine (wavelet) system generated by ψ, associated to an invertible matrix a and a lattice Γ, is the collection of functions
In this paper we prove that the set of functions generating affine systems that are a Riesz basis of
${\cal L}^2({\Bbb R}^d)$ is dense in We also prove that a stronger result is true for affine systems that are a frame of
In this case we show that the generators associated to a fixed but arbitrary dilation are a dense set. Furthermore, we analyze
the orthogonal case in which we prove that the set of generators of orthogonal (not necessarily complete) affine systems,
that are compactly supported in frequency, are dense in the unit sphere of
with the induced metric. As a byproduct we introduce the p-Grammian of a function and prove a convergence result of this
Grammian as a function of the lattice. This result gives insight in the problem of oversampling of affine systems. 相似文献
15.
A.J.E.M. Janssen 《Journal of Fourier Analysis and Applications》1994,1(4):403-436
Let
and let
In this paper we investigate the relation between the frame operator
and the matrix
whose entries
are given by
for
Here
, for any
We show that
is bounded as a mapping of
into
if and only if
is bounded as a mapping of
into
Also we show that
if and
only if
where
denotes the identity operator of
and
respectively, and
Next, when
generates a frame, we have that
has an upper frame bound, and the minimal dual function
can be computed as
The results of this paper extend, generalize, and rigourize results of Wexler and Raz and of Qian, D. Chen, K. Chen, and
Li on the computation of dual functions for finite, discrete-time Gabor expansions to the infinite, continuous-time case.
Furthermore, we present a framework in which one can show that certain smoothness and decay properties of a
generating a frame are inherited by
In particular, we show that
when
generates a frame
Schwartz space). The proofs of the main results of this paper rely heavily on a technique introduced by Tolimieri and Orr
for relating frame bound questions on complementary lattices by means of the Poisson summation formula. 相似文献
16.
In this paper we show that there exists a
-coreset for k-median and k-means clustering of n points in
which is of size independent of n. In particular, we construct a
-coreset of size
for k-median clustering, and of size
for k-means clustering. 相似文献
17.
Zoltan Furedi 《Discrete and Computational Geometry》2007,38(2):273-288
Let
be a triangle and let
be a set of homothetic copies of
. We prove that
implies that there are positive and negative signs
and there exist translates of
that cover
. 相似文献
18.
Frame Decomposition of Decomposition Spaces 总被引:3,自引:0,他引:3
A new construction of tight frames for
with flexible time-frequency localization
is considered. The frames can be adapted to form atomic decompositions for a large family of smoothness spaces on
a class of so-called decomposition spaces. The decomposition space norm can be completely characterized by a sparseness condition
on the frame coefficients. As examples of the general construction, new tight frames yielding decompositions of Besov space,
anisotropic Besov spaces, α-modulation spaces, and anisotropic α-modulation spaces are considered. Finally, curvelet-type
tight frames are constructed on
相似文献
19.
An affine pseudo-plane X is a smooth affine surface defined over
which is endowed with an
-fibration such that every fiber is irreducible and only one fiber is a multiple fiber. If there is a hyperbolic
-action on X and X is an
-surface, we shall show that the universal covering
is isomorphic to an affine hypersurface
in the affine 3-space
and X is the quotient of
by the cyclic group
via the action
where
and
It is also shown that a
-homology plane X with
and a nontrivial
-action is an affine pseudo-plane. The automorphism group
is determined in the last section. 相似文献
20.
Denote by
the real-linear span of
, where
Under the concept of left-monogeneity defined through the generalized
Cauchy-Riemann operator we obtain the direct sum decomposition of
where
is the right-Clifford module of finite linear combinations of functions of the form
, where, for
, the function R is a k- or
-homogeneous leftmonogenic
function, for
or
, respectively, and h is a function defined in [0,∞) satisfying a certain integrability condition in relation to k, the spaces
are invariant under Fourier transformation.
This extends the classical result for
. We also deduce explicit Fourier transform
formulas for functions of the form
refining Bochner’s formula for spherical k-harmonics. 相似文献