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1.
Equations with two time scales (refinement equations or dilation equations) are central to wavelet theory. Several applications
also include an inhomogeneous forcing term F(t). We develop here a part of the existence theory for the inhomogeneous refinement
equation
where a (k) is a finite sequence and F is a compactly supported distribution on ℝ.
The existence of compactly supported distributional solutions to an inhomogeneous refinement equation is characterized in
terms of conditions on the pair (a, F).
To have Lp solutions from F ∈ Lp(ℝ), we construct by the cascade algorithm a sequence of functions φ0 ∈ Lp(ℝ) from a compactly supported initial function ℝ as
A necessary and sufficient condition for the sequence {φn} to converge in Lp(ℝ)(1 ≤ p ≤ ∞) is given by the p-norm joint spectral radius of two matrices derived from the mask a. A convexity property
of the p-norm joint spectral radius (1 ≤ p ≤ ∞) is presented.
Finally, the general theory is applied to some examples and multiple refinable functions.
Acknowledgements and Notes. Research supported in part by Research Grants Council and City University of Hong Kong under Grants #9040281, 9030562, 7000741. 相似文献
2.
Song Li 《中国科学A辑(英文版)》2003,46(3):364-375
The purpose of this paper is to investigate the refinement equations of the form
where the vector of functions ϕ=(ϕ
1..., ϕ
r
)
T
is in (L
p
(ℝ
s
))
r
, 1⩽p⩽∞, a(α), α∈ℤ
s
is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that lim→∞ M-n = 0. In order to solve the refinement equation mentioned above, we start with a vector of compactly supported functions φ
0∈(L
p
(ℝ
s
))
r
and use the iteration schemes f
n
:=Q
a
n
φ
0, n=1,2,..., where Q
n
is the linear operator defined on (L
p
(ℝ
s
))
r
given by
This iteration scheme is called a subdivision scheme or cascade algorithm. In this paper, we characterize the Lp-convergence of subdivision schemes in terms of the p-norm joint spectral radius of a finite collection of some linear operators
determined by the sequence a and the set B restricted to a certain invariant subspace, where the set B is a complete set of representatives of the distinct cosets of the quotient group ℤs/Mℤs containing 0. 相似文献
3.
Ilya A. Krishtal Benjamin D. Robinson Guido L. Weiss Edward N. Wilson 《Journal of Geometric Analysis》2007,17(1):87-96
An orthonormal wavelet system in ℝd, d ∈ ℕ, is a countable collection of functions {ψ
j,k
ℓ
}, j ∈ ℤ, k ∈ ℤd, ℓ = 1,..., L, of the form
that is an orthonormal basis for L2 (ℝd), where a ∈ GLd (ℝ) is an expanding matrix. The first such system to be discovered (almost 100 years ago) is the Haar system for which L
= d = 1, ψ1(x) = ψ(x) = κ[0,1/2)(x) − κ[l/2,1)
(x), a = 2. It is a natural problem to extend these systems to higher dimensions. A simple solution is found by taking appropriate
products Φ(x1, x2, ..., xd) = φ1 (x1)φ2(x2) ... φd(xd) of functions of one variable. The obtained wavelet system is not always convenient for applications. It is desirable to
find “nonseparable” examples. One encounters certain difficulties, however, when one tries to construct such MRA wavelet systems.
For example, if a = (
1-1
1 1
) is the quincunx dilation matrix, it is well-known (see, e.g., [5]) that one can construct nonseparable Haar-type scaling
functions which are characteristic functions of rather complicated fractal-like compact sets. In this work we shall construct
considerably simpler Haar-type wavelets if we use the ideas arising from “composite dilation” wavelets. These were developed
in [7] and involve dilations by matrices that are products of the form ajb, j ∈ ℤ, where a ∈ GLd(ℝ) has some “expanding” property and b belongs to a group of matrices in GLd(ℝ) having |det b| = 1. 相似文献
4.
Xiao Ping Yuan 《数学学报(英文版)》2001,17(2):253-262
We prove the existence of quasiperiodic solutions and Lagrange stability for a class of differential equations with jumping
nonlinearity
, where a,b > 0, p(t) ∈C(ℝ/2πℤ) and φ : ℝ→ℝ is an unbounded function.
Supported by the National Natural Science Foundation of China 相似文献
5.
Camil Muscalu 《Journal of Geometric Analysis》1999,9(4):683-691
If N ∈ ℕ, 0 < p ≤ 1, and(Xk)
k=1
N
are r.i.p-spaces, it is shown that there is C(= C(p, N)) > 0, such that for every ƒ ∈ ∩
k=1
N
Xk, there exists
with
, for every 1 ≤ k ≤ N. Also, if ⊓ is a convex polygon in ℝ2, it is proved that the N-tuple (H(X1),…, H(Xn)) is K⊓-closed with respect to (X1,…, XN) in the sense of Pisier. Everything follows from Theorem 2.1, which is a general analytic partition of unity type result. 相似文献
6.
A refinable function φ(x):ℝn→ℝ or, more generally, a refinable function vector Φ(x)=[φ1(x),...,φr(x)]T is an L1 solution of a system of (vector-valued) refinement equations involving expansion by a dilation matrix A, which is an expanding
integer matrix. A refinable function vector is called orthogonal if {φj(x−α):α∈ℤn, 1≤j≤r form an orthogonal set of functions in L2(ℝn). Compactly supported orthogonal refinable functions and function vectors can be used to construct orthonormal wavelet and
multiwavelet bases of L2(ℝn). In this paper we give a comprehensive set of necessary and sufficient conditions for the orthogonality of compactly supported
refinable functions and refinable function vectors. 相似文献
7.
V. E. Slyusarchuk 《Mathematical Notes》2000,68(3):386-391
Necessary and sufficient conditions for the Lipschitzian invertibility of the difference map
, wheref: ℝ → ℝ is a continuous function, in the spacesl
p
(ℤ, ℝ), where 1≤p≤∞, of two-sided numerical sequences are obtained.
Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 448–454, September, 2000. 相似文献
8.
In this paper, we investigate compactly supported Riesz multiwavelet sequences and Riesz multiwavelet bases for L
2(ℝ
s
). Suppose ψ = (ψ1,..., ψ
r
)
T
and are two compactly supported vectors of functions in the Sobolev space (H
μ(ℝ
s
))
r
for some μ > 0. We provide a characterization for the sequences {ψ
jk
l
: l = 1,...,r, j ε ℤ, k ε ℤ
s
} and to form two Riesz sequences for L
2(ℝ
s
), where ψ
jk
l
= m
j/2ψ
l
(M
j
·−k) and , M is an s × s integer matrix such that lim
n→∞
M
−n
= 0 and m = |detM|. Furthermore, let ϕ = (ϕ1,...,ϕ
r
)
T
and be a pair of compactly supported biorthogonal refinable vectors of functions associated with the refinement masks a, and M, where a and are finitely supported sequences of r × r matrices. We obtain a general principle for characterizing vectors of functions ψν = (ψν1,...,ψνr
)
T
and , ν = 1,..., m − 1 such that two sequences {ψ
jk
νl
: ν = 1,..., m − 1, l = 1,...,r, j ε ℤ, k ε ℤ
s
} and { : ν=1,...,m−1,ℓ=1,...,r, j ∈ ℤ, k ∈ ℤ
s
} form two Riesz multiwavelet bases for L
2(ℝ
s
). The bracket product [f, g] of two vectors of functions f, g in (L
2(ℝ
s
))
r
is an indispensable tool for our characterization.
This work was supported by National Natural Science Foundation of China (Grant Nos. 10771190, 10471123) 相似文献
9.
Let
be a biquadratic number field (where d,m,n∈ℤ, are uniquely determined); we say that it is monogenic if its ring of integers
OK is of the form ℤ[θ]. We show that K is monogenic if and only if the two following conditions are satisfied:
We characterize all the imaginary monogenic biquadratic fieds and establishe other necessary conditions for monogenicity of
real fields. Conjectures, numerical tables and statistics are given. 相似文献
(i) | 2δm=2δn+4(2−δd) where δ=0 or 1 is defined by mn≡(−1)δ mod4; |
(ii) | the equation (u2-v2)2(2δm)-(u2+v2)2(2δn)=±1 has solutions in ℤ. |
10.
11.
In the paper, the equation
is considered in the scale of the weighted spaces H
β
s
(ℝ
n
) (q > 1, a
kα
∈ ℂ). We prove that if the expression
does not vanish on the set {ξ ∈ ℝ
n
∖ 0, |z| ≤ q
β−s+n
/2−2m}, then this equation has a unique solution u ∈ H
β
s+2m
(ℝ
n
) for every function f ∈ H
β
s
(ℝ
n
) provided that β, s ≠ ∈ ℝ, β − s ≠ n/2 + p, and β − s − 2m ≠ − n/2 − p (p = 0, 1, ...).
__________
Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 37–55, 2007. 相似文献
12.
E. G. Goluzina 《Journal of Mathematical Sciences》1998,89(1):958-966
Let TR be the class of functions
that are regular and typically real in the disk E={z:⋱z⋱<1}. For this class, the region of values of the system {f(z0), f(r)} for z0 ∈ ℝ, r∈(-1,1) is studied. The sets Dr={f(z0):f∈TR, f(r)=a} for −1≤r≤1 and Δr={(c2, c3): f ∈ TR, −f(−r)=a} for 0<r≤1 are found, where aε(r(1+r)−2, r(1−r)−2) is an arbitrary fixed number. Bibliography: 11 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 69–79. 相似文献
13.
Let Ω ⊂ ℝ
N
be a smooth bounded domain such that 0 ∈ Ω,N≥3, 0≤s<2,2* (s)=2(N−s)/(N−2). We prove the existence of nontrival solutions for the singular critical problem
with Dirichlet boundary condition on Ω for suitable positive parameters λ and μ.
Corresponding author. This work is supported partly by the National Natural Science Foundation of China (No. 10171036) and
the Natural Science Foundation of South-Central University For Nationalities (No. YZZ03001). The authors sincerely thank Prof.
Daomin Cao (AMSS, Chinese Academy of Sciences) for helpful discussions and suggestions. 相似文献
14.
A Gabor frame multiplier is a bounded operator that maps normalized tight Gabor frame generators to normalized tight Gabor
frame generators. While characterization of such operators is still unknown, we give a complete characterization for the functional
Gabor frame multipliers. We prove that a L∞ -function h is a functional Gabor frame multiplier (for the time-frequency lattice aℤ × bℤ) if and only if it is unimodular
and
is a-periodic. Along the same line, we also characterize all the Gabor frame generators g (resp. frame wavelets ψ) for which
there is a function ∈ L∞(ℝ) such that {wgmn} (resp. ωψk,ℝ) is a normalized tight frame. 相似文献
15.
LI Song & ZHOU Xinlong Department of Mathematics Zhejiang University Hangzhou China Instititute of Mathematics University of Duisburg-Essen D- Duisburg Germany 《中国科学A辑(英文版)》2006,49(4):439-450
This paper is concerned with multivariate refinement equations of the type where (?) is the unknown function defined on the s-dimensional Euclidean space Rs, a is a finitely supported nonnegative sequence on Zs, and M is an s×s dilation matrix with m := |detM|. We characterize the existence of L2-solution of refinement equation in terms of spectral radius of a certain finite matrix or transition operator associated with refinement mask a and dilation matrix M. For s = 1 and M = 2, the sufficient and necessary conditions are obtained to characterize the existence of continuous solution of this refinement equation. 相似文献
16.
Abdelmajid Siai 《Potential Analysis》2006,24(1):15-45
Let Ω be an open bounded set in ℝN, N≥3, with connected Lipschitz boundary ∂Ω and let a(x,ξ) be an operator of Leray–Lions type (a(⋅,∇u) is of the same type as the operator |∇u|p−2∇u, 1<p<N). If τ is the trace operator on ∂Ω, [φ] the jump across ∂Ω of a function φ defined on both sides of ∂Ω, the normal derivative
∂/∂νa related to the operator a is defined in some sense as 〈a(⋅,∇u),ν〉, the inner product in ℝN, of the trace of a(⋅,∇u) on ∂Ω with the outward normal vector field ν on ∂Ω. If β and γ are two nondecreasing continuous real functions everywhere
defined in ℝ, with β(0)=γ(0)=0, f∈L1(ℝN), g∈L1(∂Ω), we prove the existence and the uniqueness of an entropy solution u for the following problem,
in the sense that, if Tk(r)=max {−k,min (r,k)}, k>0, r∈ℝ, ∇u is the gradient by means of truncation (∇u=DTku on the set {|u|<k}) and
, u measurable; DTk(u)∈Lp(ℝN), k>0}, then
and u satisfies,
for every k>0 and every
.
Mathematics Subject Classifications (2000) 35J65, 35J70, 47J05. 相似文献
17.
Bin Han 《Advances in Computational Mathematics》2006,24(1-4):375-403
In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space Wpk(ℝs) (1≤p≤∞) to a vector refinement equation with a general dilation matrix. The criterion is constructive and can be implemented.
Rate of convergence of vector cascade algorithms in a Sobolev space Wpk(ℝs) will be investigated. When the dilation matrix is isotropic, a characterization will be given for the Lp (1≤p≤∞) critical smoothness exponent of a refinable function vector without the assumption of stability on the refinable function
vector. As a consequence, we show that if a compactly supported function vector φ∈Lp(ℝs) (φ∈C(ℝs) when p=∞) satisfies a refinement equation with a finitely supported matrix mask, then all the components of φ must belong to a Lipschitz
space Lip(ν,Lp(ℝs)) for some ν>0. This paper generalizes the results in R.Q. Jia, K.S. Lau and D.X. Zhou (J. Fourier Anal. Appl. 7 (2001) 143–167)
in the univariate setting to the multivariate setting.
Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday
Mathematics subject classifications (2000) 42C20, 41A25, 39B12.
Research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC Canada) under Grant
G121210654. 相似文献
18.
For any a,b∈R let ϕa,b(x)=ax+b(x∈R). Suppose 0<a<1. Let Ca,b be the generalized a-Cantor set with generating iterated function systme {ϕa,0, ϕa,b; ϕa,l}. Then we prove the Hausdorff dimension of
is
when 0<a≤2 cos 80°. 相似文献
19.
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
20.
We study some properties of sets of differences of dense sets in ℤ2 and ℤ3 and their interplay with Bohr neighbourhoods in ℤ. We obtain, inter alia, the following results.
(i) | If E ⊂ ℤ2, $
\bar d
$
\bar d
(E) > 0 and p
i
, q
i
∈ ℤ[x], i = 1, ..., m satisfy p
i
(0) = q
i
(0) = 0, then there exists B ⊂ ℤ such that $
\bar d
$
\bar d
(B) > 0 and
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