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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.
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
3.
Consider the diagonal action of
on the affine space
where
an algebraically closed field of characteristic
We construct a "standard monomial" basis for the ring of invariants
As a consequence, we deduce that
is Cohen-Macaulay. As the first application, we present the first and second fundamental theorems for
-actions. As the second application, assuming that the characteristic of K is
we give a characteristic-free proof of the Cohen-Macaulayness of the moduli space
of equivalence classes of semi-stable, rank 2, degree 0 vector bundles on a smooth projective curve of genus > 2. As the
third application, we describe a K-basis for the ring of invariants for the adjoint action of
on m copies of
in terms of traces. 相似文献
4.
5.
6.
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
7.
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
8.
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
9.
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
10.
Sadahiro Saeki 《Journal of Fourier Analysis and Applications》1995,2(1):15-28
Let
and
Under certain conditions on
we shall prove that
converges nontangentially to
at
for
相似文献
11.
Given a finite subset
of an additive group
such as
or
, we are interested in efficient covering of
by translates of
, and efficient packing of translates of
in
. A set
provides a covering if the translates
with
cover
(i.e., their union is
), and the covering will be efficient if
has small density in
. On the other hand, a set
will provide a packing if the translated sets
with
are mutually disjoint, and the packing is efficient if
has large density.
In the present part (I) we will derive some facts on these concepts when
, and give estimates for the minimal covering densities and maximal packing densities of finite sets
. In part (II) we will again deal with
, and study the behaviour of such densities under linear transformations. In part (III) we will turn to
.
Authors’ address: Department of Mathematics, University of Colorado at Boulder, Campus Box 395, Boulder, Colorado 80309-0395,
USA
The first author was partially supported by NSF DMS 0074531. 相似文献
12.
D.S. Lubinsky 《Constructive Approximation》2007,25(3):303-366
Assume
is not an integer. In papers published in 1913 and 1938,
S.~N.~Bernstein established the limit
Here
denotes the error in best uniform approximation of
by polynomials
of degree
. Bernstein proved that
is itself the error in best uniform approximation of
by entire functions of exponential type at most 1,
on the whole real line. We prove that the best approximating entire function
is unique, and satisfies an alternation property. We show that the scaled
polynomials of best approximation converge to this unique entire function.
We derive a representation for
, as well
as its
analogue for
. 相似文献
13.
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
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.
Jan Draisma 《Transformation Groups》2006,11(4):609-624
For a finite-dimensional representation
of a group G, the diagonal action of G on
p-tuples of elements of M, is usually poorly understood. The algorithm presented here computes a geometric characteristic
of this action in the case where G is connected and reductive, and
is a morphism of algebraic groups: The algorithm takes as input the
weight system of M, and it returns the number of irreducible components
of the null-cone of G on
for large p. The paper concludes with a theorem that if the characteristic is zero and G is semisimple, then only few M have
the property that
is small for all p. 相似文献
16.
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
17.
In this paper we develop a robust uncertainty principle for
finite signals in
which states that, for nearly all choices
such that
there is no signal
supported on
whose discrete Fourier transform
is supported on
In fact, we can make the above uncertainty principle quantitative in the sense that if
is supported on
then only a small percentage of the energy (less than half, say) of
is concentrated on
As an application of this robust uncertainty principle (QRUP), we consider the problem of decomposing a signal into a sparse
superposition of spikes and complex sinusoids
We show that if a generic signal
has a decomposition
using spike and frequency locations in
and
respectively, and obeying
then
is the unique sparsest possible decomposition (all other decompositions have more nonzero terms). In addition, if
then the sparsest
can be found by solving a convex optimization problem. Underlying our results is a new probabilistic approach which insists
on finding the correct uncertainty relation, or the optimally sparse solution for nearly all subsets but not necessarily all
of them, and allows us to considerably sharpen previously known results [9], [10]. In fact, we show that the fraction of sets
for which the above properties do not hold can be upper bounded by quantities like
for large values of
The QRUP (and the application to finding sparse representations) can be extended to general pairs of orthogonal bases
For nearly all choices
obeying
where
there is no signal
such that
is supported on
and
is supported on
where
is the mutual coherence between
and
An erratum to this article is available at . 相似文献
18.
We study the composition of the functor from the category of modules over the Lie algebra
to the category of modules over the degenerate affine Hecke algebra of GLN introduced by I. Cherednik, with the functor from the latter category to the category of modules over the Yangian
due to V. Drinfeld. We propose a representation theoretic explanation of a link between the intertwining operators on the
tensor products of
-modules, and the "extremal cocycle" on the Weyl group of
defined by D. Zhelobenko. We also establish a connection between the composition of the functors, and the "centralizer construction"
of the Yangian
discovered by G. Olshanski. 相似文献
19.
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
20.
This paper deals with the homogenization of a sequence of non-linear conductivity energies in a bounded open set
The energy density is of the same order as
where
is periodic, u is a vector-valued function in
and
The conductivity
is equal to 1 in the "hard" phases composed by
two by two disjoint-closure periodic sets while
tends uniformly to 0 in the "soft" phases composed by periodic thin layers which separate the hard phases. We prove that
the limit energy, according to γ-convergence, is a multi-phase functional equal to the sum of the homogenized energies (of
order 1) induced by the hard phases plus an interaction energy (of order 0) due to the soft phases. The number of limit phases
is less than or equal to N and is obtained by evaluating the γ-limit of the rescaled energy of density
in the torus. Therefore, the homogenization result is achieved by a double γ-convergence procedure since the cell problem
depends on ε. 相似文献