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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.
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
3.
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
4.
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
5.
Maria Roginskaya Michal Wojciechowski 《Journal of Fourier Analysis and Applications》2006,12(2):213-223
We study how the singularity (in the sense of Hausdorff dimension) of a vector valued measure can be affected by certain restrictions
imposed on its Fourier transform. The restrictions, we are interested in, concern the direction of the (vector) values of
the Fourier transform. The results obtained could be considered as a generalizations of F. and M. Riesz theorem, however a
phenomenon,
which have no analogy in the scalar case, arise in the vector valued case. As an example of application, we show that every
measure from
annihilating gradients
of
embedded in the natural way into
i.e., such that
for
, has Hausdorff dimension at least one. We provide examples which show both completeness and incompleteness of our results. 相似文献
6.
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
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.
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
9.
Alexander M. Powell 《Journal of Fourier Analysis and Applications》2005,11(4):375-387
We show that there exists an orthonormal basis
for
such that
and
are bounded
sequences. We also show that there does not exist any orthonormal basis
for
,
and
being bounded sequences.
This is motivated by a question posed by H.S. Shapiro on the mean and variance
sequences associated to orthonormal bases. 相似文献
10.
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
11.
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
12.
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
13.
Mario Petrich 《Semigroup Forum》2007,75(1):45-69
A normal cryptogroup S is a completely regular semigroup in which
is a congruence and
is a normal band. We represent S as a strong semilattice of completely simple semigroups, and may set
For each
we set
and represent
by means of an h-quintuple
These parameters are used to characterize certain quasivarieties of normal cryptogroups. Specifically, we construct the lattice
of quasivarieties generated by the (quasi)varieties
and
This is the lattice generated by the lattice of quasivarieties of normal bands, groups and completely simple semigroups.
We also determine the B-relation on the lattice of all quasivarieties of normal cryptogroups. Each quasivariety studied is
characterized in several ways. 相似文献
14.
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
15.
16.
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
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.
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
19.
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
20.
L. Evain 《Transformation Groups》2007,12(2):227-249
Let X be a smooth projective toric surface, and
the Hilbert scheme parametrizing the length d zero-dimensional subschemes of X. We compute the rational Chow ring
. More precisely, if
is the two-dimensional torus contained in X, we compute the rational equivariant Chow ring
and the usual Chow ring is an explicit quotient of the equivariant Chow ring. The case of some quasi-projective toric surfaces
such as the affine plane are described by our method too. 相似文献