共查询到20条相似文献,搜索用时 46 毫秒
1.
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
2.
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
3.
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
4.
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
5.
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
6.
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
7.
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
8.
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
9.
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
10.
This paper presents an expansion for radial tempered distributions on
in terms of smooth, radial analyzing and synthesizing functions with space-frequency localization properties similar to standard
wavelets. Scales of quasi-norms are defined for the coefficients of the expansion that characterize, via Littlewood-Paley-Stein
theory, when a radial distribution belongs to a Triebel-Lizorkin or Besov space. These spaces include, for example, the
spaces,
Hardy spaces
Sobolev spaces
and Lipschitz
spaces
We also present a smooth radial atomic decomposition and norm estimates for sums of smooth radial molecules. The radial
wavelets, atoms, and molecules that we consider are localized near certain annuli, as opposed to cubes in the usual, nonradial
setting. The radial wavelet expansion is multiscale, where the functions in the different scales are related by dilation.
However, there is no translation structure within a given scale, unlike the situation with standard wavelet systems. 相似文献
11.
Jay Rothman 《Journal of Fourier Analysis and Applications》1995,2(3):217-225
The Adler-Konheim theorem [Proc. Amer. Math. Soc. 13 (1962), 425-428] states that the collection of nth-order autocorrelation
functions
is a complete set of translation invariants for real-valued L1 functions on a locally compact abelian group. It is shown here that there are proper subsets of
that also form a complete set of translation invariants, and these subsets are characterized. Specifically, a subset is
complete if and only if it contains infinitely many even-order autocorrelation functions. In addition, any infinite subset
of
is complete up to a sign. While stated here for functions on
the proofs presented hold for functions on any locally compact abelian group that is not compact, in particular, on
and the integer lattice
相似文献
12.
Robert R. Jensen 《Journal of Fourier Analysis and Applications》1995,2(3):237-259
Let $L[\,\cdot\,]Let
be a nondivergent linear second-order uniformly elliptic partial differential operator defined on functions with domain
Consider the question, "When is a function u a solution of
on
?" The naive answer, "u is a solution of
on
if
and
for all
" is clearly too limited. Indeed, if the coefficients of L are in
then L can be rewritten in divergence form for which the notion of a "weak" solution can be applied. In this case there
could be infinitely many functions that are "weak" but not classical solutions. More importantly, even if the coefficients
of L are just bounded and measurable, the recent results of Krylov permit us to construct "solutions" of
on
and these "solutions" are generally no better than continuous; the "weak" solutions previously mentioned can be obtained
by this construction, too. The preceding discussion provides us with an adequate extrinsic definition of solution (i.e., given
a function u we either prove that it is or is not the result of such a construction) that has been used by several authors,
but one that is not particularly satisfying or illuminating. Our major contribution in this paper is to show the following.
I. There is an intrinsic definition of solution that is equivalent to the extrinsic one. II. Furthermore, the intrinsic definition
is just the (now) well-known Crandall-Lions viscosity solution, modified in a natural way to accommodate measurable coefficients. 相似文献
13.
Kernel and Trace Operators for Extensions of Brandt Semigroups 总被引:1,自引:0,他引:1
Mario Petrich 《Semigroup Forum》2007,75(1):18-44
Let S be an (ideal) extension of a Brandt semigroup S0 by a Brandt semigroup S1 and let
denote the congruence lattice of S. For
denote by
and
the least and the greatest congruences on S with the same kernel as
respectively, and let
and
have the analogous meaning relative to trace. We establish necessary and sufficient conditions on S in order that one or
more of the operators
be
- or
-homomorphisms on
The conditions are expressed directly in terms of a construction of an extension of S0 and S1 and the proofs make use of a construction of congruences on S expressed by means of congruences on S0 and S1. 相似文献
14.
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
15.
Adelheid Fischer 《Journal of Fourier Analysis and Applications》1995,2(2):161-180
In this paper we derive rates of approximation for a class of linear operators on
associated with a multiresolution analysis
We show that for a uniformly bounded sequence of linear operators
satisfying
on the subspace
a lower bound for the approximation order is determined by the number of vanishing moments of a prewavelet set. We consider
applications to extensions of generalized projection operators as well as to sampling series. 相似文献
16.
Let
be a nontrivial probability measure on the unit circle
the density of its absolutely continuous part,
its Verblunsky coefficients, and
its monic orthogonal polynomials. In this paper we compute the coefficients of
in terms of the
. If the function
is in
, we do the same for its Fourier coefficients. As an application we prove that if
and if
is a polynomial, then with
and S the left-shift operator on sequences we have
We also study relative ratio asymptotics of the reversed polynomials
and provide a necessary and sufficient condition in terms of the Verblunsky coefficients of the measures
and
for this difference to converge to zero uniformly on compact subsets of
. 相似文献
17.
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
18.
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
19.
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
20.
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