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1.
This paper generalizes the mixed extension principle in L
2(ℝ
d
) of (Ron and Shen in J. Fourier Anal. Appl. 3:617–637, 1997) to a pair of dual Sobolev spaces H
s
(ℝ
d
) and H
−s
(ℝ
d
). In terms of masks for φ,ψ
1,…,ψ
L
∈H
s
(ℝ
d
) and
, simple sufficient conditions are given to ensure that (X
s
(φ;ψ
1,…,ψ
L
),
forms a pair of dual wavelet frames in (H
s
(ℝ
d
),H
−s
(ℝ
d
)), where
For s>0, the key of this general mixed extension principle is the regularity of φ, ψ
1,…,ψ
L
, and the vanishing moments of
, while allowing
,
to be tempered distributions not in L
2(ℝ
d
) and ψ
1,…,ψ
L
to have no vanishing moments. So, the systems X
s
(φ;ψ
1,…,ψ
L
) and
may not be able to be normalized into a frame of L
2(ℝ
d
). As an example, we show that {2
j(1/2−s)
B
m
(2
j
⋅−k):j∈ℕ0,k∈ℤ} is a wavelet frame in H
s
(ℝ) for any 0<s<m−1/2, where B
m
is the B-spline of order m. This simple construction is also applied to multivariate box splines to obtain wavelet frames with short supports, noting
that it is hard to construct nonseparable multivariate wavelet frames with small supports. Applying this general mixed extension
principle, we obtain and characterize dual Riesz bases
in Sobolev spaces (H
s
(ℝ
d
),H
−s
(ℝ
d
)). For example, all interpolatory wavelet systems in (Donoho, Interpolating wavelet transform. Preprint, 1997) generated by an interpolatory refinable function φ∈H
s
(ℝ) with s>1/2 are Riesz bases of the Sobolev space H
s
(ℝ). This general mixed extension principle also naturally leads to a characterization of the Sobolev norm of a function in
terms of weighted norm of its wavelet coefficient sequence (decomposition sequence) without requiring that dual wavelet frames
should be in L
2(ℝ
d
), which is quite different from other approaches in the literature.
相似文献
2.
多尺度分析生成元的刻画 总被引:1,自引:0,他引:1
本文将给出多尺度分析生成元的一种完全刻画.将证明:函数φ∈L~2(R)是二进多尺度分析生成元的充要条件是(1)存在{a_k}∈l~2,φ(x)=∑_(k∈Z)a_kφ(2x-k);(2)存在正数A相似文献
3.
We introduce the concepts of quasi-biorthogonal frame multiresolution analyses and quasi-biorthogonal frame wavelets which are natural generalizations of biorthogonal multiresolution analyses and biorthogonal wavelets, respectively. Necessary and sufficient conditions for quasi-biorthogonal frame multiresolution analyses to admit quasi-biorthogonal wavelet frames are given, and a non-trivial example of quasi-biorthogonal frame multiresolution analyses admitting quasi-biorthogonal frame wavelets is constructed. Finally, we characterize the pair of quasi-biorthogonal frame wavelets that is associated with quasi-biorthogonal frame multiresolution analyses. 相似文献
4.
Hardin Douglas P. Hogan Thomas A. Sun Qiyu 《Advances in Computational Mathematics》2004,20(4):367-384
We use the matrix-valued Fejér–Riesz lemma for Laurent polynomials to characterize when a univariate shift-invariant space has a local orthonormal shift-invariant basis, and we apply the above characterization to study local dual frame generators, local orthonormal bases of wavelet spaces, and MRA-based affine frames. Also we provide a proof of the matrix-valued Fejér–Riesz lemma for Laurent polynomials. 相似文献
5.
四元数Hilbert空间在应用物理科学特别是量子物理中占有重要地位.本文讨论四元数Hilbert空间的框架理论, 在四元数Hilbert空间中引入了Riesz基的概念, 在此基础上刻画了Riesz基,给出了它们的一些等价条件; 特别地, 得到了四元数Hilbert空间中的一个序列是Riesz基的充要条件是它是一个具有双正交序列的完备Bessel序列,且它的双正交序列也是一个完备Bessel序列; 并进一步证明了双正交序列中一个序列的完备性可以从特征刻画中去除.文中举例说明了双正交性、完备性和Bessel性质之间的关系. 相似文献
6.
Bin Han 《Proceedings of the American Mathematical Society》2006,134(7):1973-1983
Let be a compactly supported refinable function in such that the shifts of are stable and for a -periodic trigonometric polynomial . A wavelet function can be derived from by . If is an orthogonal refinable function, then it is well known that generates an orthonormal wavelet basis in . Recently, it has been shown in the literature that if is a -spline or pseudo-spline refinable function, then always generates a Riesz wavelet basis in . It was an open problem whether can always generate a Riesz wavelet basis in for any compactly supported refinable function in with stable shifts. In this paper, we settle this problem by proving that for a family of arbitrarily smooth refinable functions with stable shifts, the derived wavelet function does not generate a Riesz wavelet basis in . Our proof is based on some necessary and sufficient conditions on the -periodic functions and in such that the wavelet function , defined by , generates a Riesz wavelet basis in .
7.
The Marcinkiewicz-Zygmund inequality and the Bernstein inequality are established on ∮2m(T,R)∩L2(R) which is the space of polynomial splines with irregularly distributed nodes T={tj}j∈Z, where {tj}j∈Z is a real sequence such that {eitξ}j∈Z constitutes a Riesz basis for L2([-π,π]). From these results, the asymptotic relation E(f,Bπ,2)2=lim E(f,∮2m(T,R)∩L2(R))2 is proved, where Bπ,2 denotes the set of all functions from L2(R) which can be continued to entire functions of exponential type ≤π, i.e. the classical Paley-Wiener class. 相似文献
8.
设G是局部紧Abel群,Ω■G是Haar可测集,L~2(Ω)是Ω上Haar平方可积函数构成的Hilbert空间,PWΩ(G):={f∈L~2(G):supp f(ξ)■Ω}是G上的Paley-Wiener空间.本文研究Paley-Wiener空间PWΩ(G)上平移Riesz基和Riesz谱集Ω之间的关系. 相似文献
9.
Let S be a bounded, Riemann measurable set in Rd, and Λ be a lattice. By a theorem of Fuglede, if S tiles Rd with translation set Λ, then S has an orthogonal basis of exponentials. We show that, under the more general condition that S multi-tiles Rd with translation set Λ, S has a Riesz basis of exponentials. The proof is based on Meyer?s quasicrystals. 相似文献
10.
姚喜妍 《应用泛函分析学报》2008,10(2):155-161
引入并研究了Banach空间X中的Bessel集、广义框架与广义Riesz基.对X中的任一Bessel集{gm}m∈M,定义有界线性算子T:L^2(P)→X^*,利用算子丁,给出了Bessel集与广义框架的等价刻画.同时讨论了广义框架和广义Riesz基的摄动. 相似文献
11.
12.
R. A. Zalik 《Proceedings of the American Mathematical Society》2007,135(3):787-793
We investigate the properties of univariate MRA Riesz wavelets. In particular we obtain a generalization to semiorthogonal MRA wavelets of a well-known representation theorem for orthonormal MRA wavelets.
13.
Conditions for a system of contractions and translations of a function to be a Riesz basis are given. 相似文献
14.
Kjetil Røysland 《Journal of Fourier Analysis and Applications》2008,14(2):267-285
We give an equivariant version of Packer and Rieffel’s theorem on sufficient conditions for the existence of orthonormal wavelets
in projective multiresolution analysis. Suppose that the scaling functions are invariant with respect to some finite group
action. We give sufficient conditions for the existence of wavelets with similar invariance.
Research supported in part by the Research Council of Norway, project number NFR 154077/420. Some of the final work was also
done with the support from the project NFR 170620/V30. 相似文献
15.
Semi-fine limits at infinity are studied for Riesz potentials of functions on R
n
with a certain growth condition. We are also concerned with monotone BLD functions. 相似文献
16.
具有矩阵伸缩的双正交小波基 总被引:5,自引:0,他引:5
在这篇文章里,我们研究了伸缩为矩阵的双正交小波基的构造问题,在适当条件下,我们得到了L~2(R~n)的小波框架或双正交小波基{sj,k}和{sj,k},其中sjk(x)=detAs(Ajx-k),sj,k(x)=detAj2s(Ajx-k)(j Z. k Z~n)及 A是一伸缩矩阵. 相似文献
17.
本文说明Banach空间上p-fusion框架和p-框架有紧密联系.应用分析算子和合成算子给出p-fusion Bessel序列、p-fusion框架和q-fusion Riesz基的等价描述. 相似文献
18.
In a recent investigation [8] concerning the asymptotic behavior of Gram—Schmidt orthonormalization procedure applied to
the nonnegative integer shifts of a given function, the problem of determining whether or not such functions form a Riesz
system in arose. In this paper, we provide a sufficient condition to determine whether the nonnegative translates form a Riesz system
on . This result is applied to identify a large class of functions for which very general translates enjoy the Riesz basis property
in .
August 5, 1998. Date revised: August 25, 1999. Date accepted: January 11, 2000. 相似文献
19.
One of the best ways of studying ordered algebraic structures is through their spectra. The three well-known spectra usually considered are the Brumfiel, Keimel, and the maximal spectra. The pointfree versions of these spectra were studied by B. Banaschewski for f-rings. Here, we give the pointfree versions of the Keimel and the maximal spectra for Riesz spaces. Moreover, we briefly mention how one can use the results of this paper to give a pointfree version of the Kakutani duality for Riesz spaces. 相似文献
20.
Nico M. Temme 《Journal of Computational and Applied Mathematics》2009,232(2):201-215
Riesz fractional derivatives of a function, (also called Riesz potentials), are defined as fractional powers of the Laplacian. Asymptotic expansions for large x are computed for the Riesz fractional derivatives of the Airy function of the first kind, Ai(x), and the Scorer function, Gi(x). Reduction formulas are provided that allow one to express Riesz potentials of products of Airy functions, and , via and . Here Bi(x) is the Airy function of the second type. Integral representations are presented for the function A2(a,b;x)=Ai(x−a)Ai(x−b) with a,b∈R and its Hilbert transform. Combined with the above asymptotic expansions they can be used for computing asymptotics of the Hankel transform of . These results are used for obtaining the weak rotation approximation for the Ostrovsky equation (asymptotics of the fundamental solution of the linearized Cauchy problem as the rotation parameter tends to zero). 相似文献