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1.
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
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.
Regular Semigroups with Inverse Transversals 总被引:2,自引:0,他引:2
Fenglin Zhu 《Semigroup Forum》2006,73(2):207-218
Let C be a semiband with an inverse transversal
. In [7], G.T. Song and F.L. Zhu construct a fundamental regular semigroup
with an inverse transversal
.
is isomorphic to a subsemigroup of the Hall semigroup of C but it is easier to handle. Its elements are partial transformations,
and the operation-although not the usual composition-is defined by means of composition. Any full regular subsemigroup T of
is a fundamental regular semigroup with inverse transversal
. Moreover, any regular semigroup S with an inverse transversal
is proved to be an idempotent-separating coextension of a full regular subsemigroup T of some
. By means of a full
regular subsemigroup T of some
and by means of an inverse semigroup K satisfying some conditions, in this paper, we construct a regular semigroup
with inverse transversal
such that
is isomorphic to K and
to T. Furthermore, it is proved that if S is a regular semigroup with an inverse transversal
then S can be constructed from the corresponding T and from
in this way. 相似文献
4.
We show that every function in the Hardy space can be approximated by linear combinations of translates and dilates of a synthesizer
, provided only that
and
satisfies a mild regularity condition. Explicitly, we prove scale averaged approximation for each
,
where
is an arbitrary lacunary sequence (such as
) and the coefficients
are local averages of f. This formula holds in particular if the synthesizer
is in the Schwartz class, or if it has compact support and belongs to
for some
in terms of differences of
. 相似文献
5.
In this article we show that the distributional point values of a tempered distribution are characterized by their Fourier
transforms in the following way: If
and
, and
is locally integrable, then
distributionally if and only if there exists k such that
, for each a > 0, and similarly in the case when
is a general distribution. Here
means in the Cesaro sense. This result generalizes the characterization of Fourier series of distributions with a distributional
point value given in [5] by
. We also show that under some extra conditions, as if the sequence
belongs to the space
for some
and the tails satisfy the estimate
,\ as
, the asymmetric partial sums\ converge to
. We give convergence results in other cases and we also consider the convergence of the asymmetric partial integrals. We
apply these results to lacunary Fourier series of distributions. 相似文献
6.
Rostom Getsadze 《Journal of Fourier Analysis and Applications》2006,12(5):597-604
We prove the following theorem: For arbitrary
there exists a nonnegative
function
such that
and
almost everywhere on
where
is the double Walsh-Paley system.
This statement remains true also for the double trigonometric system. 相似文献
7.
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
8.
David Walnut 《Journal of Fourier Analysis and Applications》1995,2(5):435-452
It is shown that a function
is completely determined by the samples of
on sets
where
and
is irrational if
and of
If
then the samples of
on
and only the first k derivatives of
at 0 are required to determine f completely. Higher dimensional analogues of these results, which apply to functions
and
are proven. The sampling results are sharp in the sense that if any condition is omitted, there exist nonzero
and
satisfying the rest. It is shown that the one-dimensional sampling sets correspond to Bessel sequences of complex exponentials
that are not Riesz bases for
A signal processing application in which such sampling sets arise naturally is described in detail. 相似文献
9.
Ingrid Daubechies H.J. Landau Zeph Landau 《Journal of Fourier Analysis and Applications》1994,1(4):437-478
Gabor time-frequency lattices are sets of functions of the form
generated from a given function
by discrete translations in time and frequency. They are potential tools for the decomposition and handling of signals that,
like speech or music, seem over short intervals to have well-defined frequencies that, however, change with time. It was recently
observed that the behavior of a lattice
can be connected to that of a dual lattice
Here we establish this interesting relationship and study its properties. We then clarify the results by applying the theory
of von Neumann algebras. One outcome is a simple proof that for
to span
the lattice
must have at least unit density. Finally, we exploit the connection between the two lattices to construct expansions having
improved convergence and localization properties. 相似文献
10.
Michael I. Ganzburg 《Constructive Approximation》2008,27(3):289-321
Let B be a closed linear subspace of a Banach space F and let
be a group of continuous linear operators
, where G is a compact topological group. We prove that if
is invariant under
, then under some conditions on f, F, B, and G, there exists an element
of best approximation to f that has the same property. As applications, we compute the bivariate Bernstein constant for
polynomial approximation of
and solve a Braess problem on the exponential order of decay of the error of polynomial approximation of
. Other examples and
applications are discussed as well. 相似文献
11.
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
12.
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
13.
António M. Caetano Hans-Gerd Leopold 《Journal of Fourier Analysis and Applications》2006,12(4):427-445
The concept of local growth envelope
of the quasi-normed function space
is applied to the Triebel-Lizorkin spaces of generalized smoothness
In order to achieve this, a standardization result for these and corresponding Besov spaces is derived. 相似文献
14.
We continue the investigation of some problems in learning theory in the setting formulated by F. Cucker and S. Smale. The
goal is to find an estimator
on the base of given data
that approximates well the regression function
of an unknown Borel probability measure
defined on
We assume that
belongs to a function class
It is known from previous works that the behavior of the entropy numbers
of
in the uniform norm
plays an important role in the above problem. The standard way of measuring the error between a target function
and an estimator
is to use the
norm (
is the marginal probability measure on X generated by
). This method has been used in previous papers. We continue to use this method in this paper. The use of the
norm in measuring the error has motivated us to study the case when we make an assumption on the entropy numbers
of
in the
norm. This is the main new ingredient of thispaper. We construct good estimators in different settings: (1) we know both
and
; (2) we know
but we do not know
and (3) we only know that
is from a known collection of classes but we do not know
An estimator from the third setting is called a universal estimator. 相似文献
15.
This paper studies a class of linear operators on spaces of functions of one real variable, which correspond to multiplication
by a measurable function under the Weil transform
These operators are called Weil multipliers, and arise out of the authors' study of Gabor series and radar ambiguity functions.
Representation theory provides a natural class of Weil multipliers: the set of doubly periodic functions with absolutely convergent
Fourier series,
It will be proved that functions in
are
multipliers for all
and, therefore, define bounded linear endomorphisms of
Also, we record the fact that the Wiener lemma tells us something about the orbit structure of these multipliers acting
on function spaces on the Heisenberg nilmanifold. Linear maps that correspond to multiplication by a function under a unitary
conjugacy have a particularly simple spectral decomposition, which yields an approximation theory for these operators and
provides insight into the foundation of the authors' previous work on approximate orthonormal bases. Finally, the problem
of inversion of a multiplier will be analyzed for smooth functions that have a specified structure near their zeros. 相似文献
16.
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
17.
The goal of this paper is to extend some previous results on abelian ideals of Borel subalgebras to so-called spherical ideals of
These are ideals
of
such that their G-saturation
is a spherical G-variety. We classify all maximal spherical ideals of
for all simple G.Received: 25 March 2004 相似文献
18.
The central result of this paper is a sandwiching theorem for semigroups acting on Banach lattices with order continuous norm.
As a preparation we show that the norm of a Banach lattice
is order continuous if and only if every order bounded weak null sequence in
is a norm null sequence. From the sandwiching result we deduce approximation formulas for the modulus semigroup and its generator.
For example,
if
generates a dominated
-semigroup
we show that
converges to the modulus semigroup
of
as
, and
converges (in the strong resolvent sense) to the generator of the modulus semigroup of
as
. 相似文献
19.
A. Askari Hemmat Jean-Pierre Gabardo 《Journal of Fourier Analysis and Applications》2007,13(5):589-606
Given an invertible
matrix B and
a finite or countable subset of
, we consider the collection
generating the closed subspace
of
. If that collection forms a frame for
, one can introduce two different types of shift-generated (SG) dual frames for X, called type I and type II SG-duals, respectively.
The main distinction between them is that a SG-dual of type I is required to be contained in the space
generated by the original frame while, for a type II SG-dual, one imposes that the range of the frame transform associated
with the dual be contained in the range of the frame transform associated with the original frame. We characterize the uniqueness
of both types of duals using the Gramian and dual Gramian operators which were introduced in an article by Ron and Shen and
are known to play an important role in the theory of shift-invariant spaces. 相似文献
20.
Arthur D. Grainger 《Semigroup Forum》2006,73(2):234-242
Let J be an infinite set and let
, i.e., I is the collection of all non empty finite subsets of
J. Let
denote the collection of all ultrafilters on the set I and let
be the compact (Hausdorff) right topological semigroup that is the Stone-Cech Compactification of the semigroup
equipped with the discrete topology. This paper continues the study of
that was started in [3] and [5]. In [5], Koppelberg established that
(where K( S) is the smallest ideal of a semigroup S) and for non empty
she established
. In this note, we show that for
such that
is infinite,
is a proper subset of
and
, where
. 相似文献