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Boris Rubin 《Journal of Fourier Analysis and Applications》1998,4(2):175-197
The generalized Calderón reproducing formula involving “wavelet measure” is established for functions f ∈ Lp(ℝn). The special choice of the wavelet measure in the reproducing formula gives rise to the continuous decomposition of f into
wavelets, and enables one to obtain inversion formulae for generalized windowed X-ray transforms, the Radon transform, and
k-plane transforms. The admissibility conditions for the wavelet measure μ are presented in terms of μ itself and in terms
of the Fourier transform of μ.
Acknowledgements and Notes. Partially sponsored by the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation
(Germany). 相似文献
3.
On the spectra of a Cantor measure 总被引:1,自引:0,他引:1
We analyze all orthonormal bases of exponentials on the Cantor set defined by Jorgensen and Pedersen in J. Anal. Math. 75 (1998) 185-228. A complete characterization for all maximal sets of orthogonal exponentials is obtained by establishing a one-to-one correspondence with the spectral labelings of the infinite binary tree. With the help of this characterization we obtain a sufficient condition for a spectral labeling to generate a spectrum (an orthonormal basis). This result not only provides us an easy and efficient way to construct various of new spectra for the Cantor measure but also extends many previous results in the literature. In fact, most known examples of orthonormal bases of exponentials correspond to spectral labelings satisfying this sufficient condition. We also obtain two new conditions for a labeling tree to generate a spectrum when other digits (digits not necessarily in {0,1,2,3}) are used in the base 4 expansion of integers and when bad branches are allowed in the spectral labeling. These new conditions yield new examples of spectra and in particular lead to a surprizing example which shows that a maximal set of orthogonal exponentials is not necessarily an orthonormal basis. 相似文献
4.
We prove a Calderón reproducing formula for a continuous wavelet transform associated with a class of singular differential
operators on the half line. We apply this result to derive a new inversion formula for the generalized Abel transform. 相似文献
5.
Periodization and sampling operators are defined, and the Fourier transform of periodization is uniform sampling in a well-defined
sense. Implementing this point of view, Poisson Summation Formulas are proved in several spaces including integrable functions
of bounded variation (where the result is known) and elements of mixed norm spaces. These Poisson Summation Formulas can be
used to prove corresponding sampling theorems.
The sampling operators used to understand and prove the aforementioned Poisson Summation Formulas lead to the introduction
of spaces of continuous linear operators which commute with integer translations. Operators L of this type are appropriately
called sampling multipliers. For a given function f, they give rise to new sampling formulas, whose sampling coefficients
are of the form Lf. In practice, Lf can be used to model noisy data or data where point values are not available. By representation
theorems of the second named author, some of these operator spaces are proved to be mixed norm spaces.
The approach and results of this paper were developed in the context of Duffin and Schaeffer’s theory of frames. In particular,
sampling multipliers L are related to the Bessel map used by Duffin and Schaeffer in their definition of the frame operator.
The first named author was supported in part by AFOSR contract F49620-96-1-0193.
The second named author was supported by the Cusanuswerk. 相似文献
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Dedicated to the memory of S.K. Pichorides 相似文献
9.
On the stability of wavelet and Gabor frames (Riesz bases) 总被引:6,自引:0,他引:6
Zhang Jing 《Journal of Fourier Analysis and Applications》1999,5(1):105-125
If the sequence of functions j, k is a wavelet frame (Riesz basis) or Gabor frame (Riesz basis), we obtain its perturbation system j,k which is still a frame (Riesz basis) under very mild conditions. For example, we do not need to know that the support of or
is compact as in [14]. We also discuss the stability of irregular sampling problems. In order to arrive at some of our results, we set up a general multivariate version of Littlewood-Paley type inequality which was originally considered by Lemarié and Meyer [17], then by Chui and Shi [9], and Long [16]. 相似文献
10.
Braverman and Kazhdan(2000) introduced influential conjectures aimed at generalizing the Fourier transform and the Poisson summation formula. Their conjectures should imply that quite general Langlands L-functions have meromorphic continuations and functional equations as predicted by Langlands' functoriality conjecture. As an evidence for their conjectures, Braverman and Kazhdan(2002) considered a setting related to the so-called doubling method in a later paper and proved the corresponding Poisson summation formula under restrictive assumptions on the functions involved. The connection between the two papers is made explicit in the work of Li(2018). In this paper, we consider a special case of the setting in Braverman and Kazhdan's later paper and prove a refined Poisson summation formula that eliminates the restrictive assumptions of that paper. Along the way we provide analytic control on the Schwartz space we construct; this analytic control was conjectured to hold(in a slightly different setting) in the work of Braverman and Kazhdan(2002). 相似文献
11.
We give a recursive algorithm to compute the multivariable Zassenhaus formula e^X1+X2+…+Xn=e^X1eX2…e^Xn∏∞k=2e^Wk and derive an effective recursion formula of Wk. 相似文献
12.
Dmitry Faifman 《Comptes Rendus Mathematique》2010,348(7-8):407-410
We show that, under certain conditions, the Fourier transform is completely characterized by Poisson's summation formula. Also, we propose a generalized transform which is derived from a Poisson-type summation formula, that we call a Fourier–Poisson transform. 相似文献
13.
M. Kovalyov 《Journal of Differential Equations》2005,213(1):1-80
Practically every book on the Inverse Scattering Transform method for solving the Cauchy problem for KdV and other integrable systems refers to this method as nonlinear Fourier transform. If this is indeed so, the method should lead to a nonlinear analogue of the Fourier expansion formula . In this paper a special class of solutions of KdV whose role is similar to that of ei(kx-ω(k)t) is discussed. The theory of these solutions, referred to here as harmonic breathers, is developed and it is shown that these solutions may be used to construct more general solutions of KdV similarly to how the functions ei(kx-ω(t)) are used to perform the same task in the theory of Fourier transform. A nonlinear superposition formula for general solutions of KdV similar to the Fourier expansion formula is conjectured. 相似文献
14.
Chunjie Wang 《Integral Equations and Operator Theory》2007,58(4):597-601
We prove a stability theorem for the nullity of a linear combination c
1
P
1 + c
2
P
2 of two idempotent operators P
1, P
2 on a Banach space provided c
1, c
2 and c
1 + c
2 are nonzero. We then show that for c
1
P
1 + c
2
P
2 the property of being upper semi-Fredholm, lower semi-Fredholm and Fredholm, respectively, is independent of the choice of
c
1, c
2, and that the nullity, defect and index of c
1
P
1 + c
2
P
2 are stable. 相似文献
15.
By establishing a cosine analogue of a result of Askey and Steinig on a monotonic sine sum, this paper sharpens and unifies several results associated with Young's inequality for the partial sums of k
–1 cosk. 相似文献
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Two versions of Rubio de Francia’s extrapolation theorem for multivariable operators of functions are obtained. One version
assumes an initial estimate with different weights in each space and implies boundedness on all products of Lebesgue spaces.
Another version assumes an initial estimate with the same weight but yields boundedness on a product of Lebesgue spaces whose
indices lie on a line. Applications are given in the context of multilinear Calderón-Zygmund operators. Vector-valued inequalities
are automatically obtained for them without developing a multilinear Banach-valued theory. A multilinear extension of the
Marcinkiewicz and Zygmund theorem on ℓ2-valued extensions of bounded linear operators is also obtained. 相似文献
18.
M. Bakonyi L. Rodman I. M. Spitkovsky H. J. Woerdeman 《Journal of Fourier Analysis and Applications》1999,5(1):21-44
Let S be a band in Z2 bordered by two parallel lines that are of equal distance to the origin. Given a positive definite 1 sequence of matrices {cj}jS we prove that there is a positive definite matrix function f in the Wiener algebra on the bitorus such that the Fourier coefficients
equal ck for k S. A parameterization is obtained for the set of all positive extensions f of {cj}jS. We also prove that among all matrix functions with these properties, there exists a distinguished one that maximizes the entropy. A formula is given for this distinguished matrix function. The results are interpreted in the context of spectral estimation of ARMA processes. 相似文献
19.
Boris Rubin 《Journal of Fourier Analysis and Applications》2000,6(2):185-205
Approximate and explicit inversion formulas are obtained for a new class of exponential k-plane transforms defined by
where x∈ℝn, Θ is a k-frame in ℝn, 1≤k≤n−1, μ∈ℂk is an arbitrary complex vector. The case k=1, μ∈ℝ corresponds to the exponential X-ray transform arising in single photon
emission tomography. Similar inversion formulas are established for the accompanying transform
where V is a real (n×k)-matrix. Two alternative methods, leading to the relevant continuous wavelet transforms, are presented.
The first one is based on the use of the generalized Calderón reproducing formula and multidimensional fractional integrals
with a Bessel function in the kernel. The second method employs interrelation between Pμ and the associated oscillatory potentials. 相似文献
20.
Herbert H. H. Homeier 《Numerical Algorithms》1992,3(1):245-254
A nonlinear sequence transformation is presented which is able to accelerate the convergence of Fourier series. It is tailored to be exact for a certain model sequence. As in the case of the Levin transformation and other transformations of Levin-type, in this model sequence the partial sum of the series is written as the sum of the limit (or antilimit) and a certain remainder, i.e., it is of Levin-type. The remainder is assumed to be the product of a remainder estimate and the sum of the first terms oftwo Poincaré-type expansions which are premultiplied by two different phase factors. This occurrence of two phase factors is the essential difference to the Levin transformation. The model sequence for the new transformation may also be regarded as a special case of a model sequence based on several remainder estimates leading to the generalized Richardson extrapolation process introduced by Sidi. An algorithm for the recursive computation of the new transformation is presented. This algorithm can be implemented using only two one-dimensional arrays. It is proved that the sequence transformation is exact for Fourier series of geometric type which have coefficients proportional to the powers of a numberq, |q|<1. It is shown that under certain conditions the algorithm indeed accelerates convergence, and the order of the convergence is estimated. Finally, numerical test data are presented which show that in many cases the new sequence transformation is more powerful than Wynn's epsilon algorithm if the remainder estimates are properly chosen. However, it should be noted that in the vicinity of singularities of the Fourier series the new sequence transformation shows a larger tendency to numerical instability than the epsilon algorithm. 相似文献