共查询到20条相似文献,搜索用时 15 毫秒
1.
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
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.
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
4.
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
5.
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
6.
The interassociates of the free commutative semigroup on n generators, for n > 1, are identified. For fixed n, let (S, ·)
denote this semigroup. We show that every interassociate can be written in the form
, depending only on a n-tuple
. Next, if
and
are isomorphic interassociates of (S, ·) such that
, for xii and xj in the generating set of S, then
. Moreover,
if and only if
is a permutation of
. 相似文献
7.
We provide a direct computational proof of the known inclusion
where
is the product Hardy space defined for example by R. Fefferman and
is the classical Hardy space used, for example, by E.M. Stein. We
introduce a third space
of Hardy type and analyze the interrelations among these spaces. We give simple sufficient conditions for a given function
of two variables to be the double Fourier transform of a function in
and
respectively. In particular, we obtain a broad class of multipliers on
and
respectively. We also present analogous sufficient conditions in the case of double trigonometric series and, as a by-product,
obtain new multipliers on
and
respectively. 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.
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
10.
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
11.
Zachary Mesyan 《Semigroup Forum》2007,75(3):648-675
Let
be a countably infinite set,
the group of permutations of
, and
the monoid of self-maps of
. Given two subgroups
, let us write
if there exists a finite subset
such that the groups generated by
and
are equal. Bergman and Shelah showed that the subgroups which are closed in the function topology on S fall into exactly
four equivalence classes with respect to
. Letting
denote the obvious analog of
for submonoids of E, we prove an analogous result for a certain class of submonoids of E, from which the theorem for groups
can be recovered. Along the way, we show that given two subgroups
which are closed in the function topology on S, we have
if and only if
(as submonoids of E), and that
for every subgroup
(where
denotes the closure of G in the function topology in S and
its closure in the function topology in E). 相似文献
12.
13.
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
14.
Jesus Jeronimo Castro 《Discrete and Computational Geometry》2007,37(3):409-417
Let
be a family of convex figures in the plane. We say that
has property T if there exists a line intersecting every member of
. Also, the family
has property T(k) if every k-membered subfamily of
has property T. Let B be the unit disc centered at the origin. In this paper we prove that if a finite family
of translates of B has property T(4) then the family
, where
, has property T. We also give some results concerning families of translates of the unit disc which has either property T(3)
or property T(5). 相似文献
15.
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
16.
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
17.
For any fixed
we construct an orthonormal Schauder basis
for C[-1,1] consisting of algebraic polynomials
with
The orthogonality is with respect to the Chebyshev weight. 相似文献
18.
The aim of this paper is to study the well-posedness of the initial-boundary value problem
where
is a bounded regular open domain in
is the outward normal to
and
, where
are pairwise disjoint measurable subsets of
with respect to Lebesgue surface measure on
. The main novelty lies on the reactive dynamical boundary condition imposed on
. The technique makes it possible to study the more general initial-boundary value problem
where
is as before and
. A key step in our analysis consists in studying the eigenvalue problem
相似文献
19.
Jacek Dziubanski 《Constructive Approximation》2008,27(3):269-287
Let
be the standard Laguerre functions of type a. We denote
. Let
and
be the semigroups associated with the orthonormal systems
and
. We say that a function f belongs to the Hardy space
associated with one of the semigroups if the corresponding maximal function belongs to
. We prove special atomic decompositions of the elements of the Hardy spaces. 相似文献
20.
Radu Balan Peter G. Casazza Christopher Heil Zeph Landau 《Journal of Fourier Analysis and Applications》2006,12(2):105-143
Frames have applications in numerous fields of mathematics and engineering. The fundamental property of frames which makes
them so useful is their overcompleteness. In most applications, it is this overcompleteness that is exploited to yield a decomposition
that is more stable, more robust, or more compact than is possible using nonredundant systems. This work presents a quantitative
framework for describing the overcompleteness of frames. It introduces notions of localization and approximation between two
frames
and
(
a discrete
abelian group), relating the decay of the expansion of the elements of
in terms of the elements of
via a map
. A fundamental set of equalities are shown between three seemingly unrelated quantities: The relative measure of
, the relative measure of
— both of which are determined by certain averages of inner products of frame elements with their corresponding dual frame
elements — and the density of the set
in
. Fundamental new results are obtained on the excess and overcompleteness of frames, on the relationship between frame bounds
and density, and on the structure of the dual frame of a localized frame. In a subsequent article, these results are applied
to the case of Gabor frames, producing an array of new results as well as clarifying the meaning of existing results. The
notion of localization and related approximation properties introduced in this article are a spectrum of ideas that quantify
the degree to which elements of one frame can be approximated by elements of another frame. A comprehensive examination of
the interrelations among these localization and approximation concepts is presented. 相似文献