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1.
Multivariate Refinement Equations and Convergence of Cascade Algorithms in Lp（0〈p〈1）Spaces 总被引：1，自引：0，他引：1
SongLI 《数学学报(英文版)》2003,19(1):97-106
We consider the solutions of refinement equations written in the form
where the vector of functions ϕ = (ϕ
1, ..., ϕ
r
)
T
is unknown, g is a given vector of compactly supported functions on ℝ
s
, a is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s dilation matrix with m = |detM|. Inhomogeneous refinement equations appear in the construction of multiwavelets and the constructions of wavelets on a finite
interval. The cascade algorithm with mask a, g, and dilation M generates a sequence ϕ
n
, n = 1, 2, ..., by the iterative process
from a starting vector of function ϕ
0. We characterize the L
p
-convergence (0 < p < 1) of the cascade algorithm in terms of the p-norm joint spectral radius of a collection of linear operators associated with the refinement mask. We also obtain a smoothness
property of the solutions of the refinement equations associated with the homogeneous refinement equation.
This project is supported by the NSF of China under Grant No. 10071071 相似文献
2.
Song Li 《Advances in Computational Mathematics》2004,20(4):311-331
This paper concerns multivariate homogeneous refinement equations of the form
and multivariate nonhomogeneous refinement equations of the form
where =(1,...,
r
)T is the unknown, M is an s×s dilation matrix with m=|detM|, g=(g
1,...,g
r
)T is a given compactly supported vector-valued function on R
s
, and a is a finitely supported refinement mask such that each a() is an r×r (complex) matrix. In this paper, we characterize the optimal smoothness of a multiple refinable function associated with homogeneous refinement equations in terms of the spectral radius of the corresponding transition operator restricted to a suitable finite-dimensional invariant subspace when M is an isotropic dilation matrix. Nonhomogeneous refinement equations naturally occur in multi-wavelets constructions. Let 0 be an initial vector of functions in the Sobolev space (W
2
k
(R
s
))
r
(kN). The corresponding cascade algorithm is given by
相似文献
3.
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
4.
Song LI~ 《中国科学A辑(英文版)》2007,50(7):1015-1025
Biorthogonal multiple wavelets are generated from refinable function vectors by using the multiresolution analysis.In this paper we provide a general method for the construction of compactly supported biorthogonal multiple wavelets by refinable function vectors which are the solutions of vector refinement equations of the form (?)(x)=(?)a(α)(?)(Mx-α),x∈R~s, where the vector of functions(?)=((?)_1,...,(?)_r)~T is in(L_2(R~s))~r,a=:(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_(n→∞)M~(-n)=0.Our characterizations are in the general setting and the main results of this paper are the real extensions of some known results. 相似文献
5.
This paper is devoted to investigating the solutions of refinement equations of the form Ф（x）=∑α∈Z^s α（α）Ф（Mx-α）,x∈R^s,where the vector of functions Ф = （Ф1,… ,Фr）^T is in （L1（R^s））^r, α =（α（α））α∈Z^s is an infinitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that lim n→∞ M^-n =0, with m = detM. Some properties about the solutions of refinement equations axe obtained. 相似文献
6.
We investigate limiting behavior as γ tends to ∞ of the best polynomial approximations in the Sobolev-Laguerre space WN,2([0, ∞); e−x) and the Sobolev-Legendre space WN,2([−1, 1]) with respect to the Sobolev-Laguerre inner product
and with respect to the Sobolev-Legendre inner product
respectively, where a0 = 1, ak ≥0, 1 ≤k ≤N −1, γ > 0, and N ≥1 is an integer. 相似文献
7.
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
8.
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
9.
A. V. Filinovskii 《Mathematical Notes》1997,61(5):635-643
The following boundary value problem is studied:
here the surface Г satisfies the condition(
, where
and ν is the outward (with respect to Ω) normal to Γ.
Translated fromMatematischskie Zametki, Vol. 61, No. 5, pp. 759–768, May, 1997.
Translated by N. K. Kulman 相似文献
10.
I. E. Chyzhykov 《Ukrainian Mathematical Journal》2007,59(7):1088-1109
For arbitrary 0 ≤ σ ≤ ρ ≤ σ + 1, we describe the class A
σ
ρ
of functions g(z) analytic in the unit disk
= {z : ∣z∣ < 1} and such that g(z) ≠ 0, ρT[g] = σ, and ρM[g] = ρ, where
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 7, pp. 979–995, July, 2007. 相似文献
11.
Xiaojing Yang 《Archiv der Mathematik》2005,85(5):460-469
In this paper, the existence of unbounded solutions for the following nonlinear asymmetric oscillator
is discussed, where α, β are positive constants satisfying
for some ω ∈R+ /Q, h(t) ∈L∞ [0, 2π ] is 2π-periodic, x±=max {±x, 0 }.
Received: 23 September 2004 相似文献
12.
The asymptotic expressions of the covariance matrices for both the least square estimates
L
α
T
and Markov (best linear) estimates
are obtained, based on a sample in a finite interval (0, T) of the regression co-efficients α = (α
1, …, α
m
0)′ of a parameter-continuous process with a stationary residual. We assume that the regression variables φ
ν(t), t ⩾ 0, ν = 1, …, m
0, are continuous in t, and satisfy conditions (3.1)–(3.3). For the residual, we assume that it is a stationary process that possesses a bounded
continuous spectral density f(λ). Under these assumptions, it is proven that
where the matrices D
T
, B(0), α(λ) are defined in Section 3.
Under the assumptions mentioned above, if, furthermore, there exist some positive integer m and a constant C such that g(λ)(1 + λ
2)m ⩾ C > 0, where g(λ) is the spectral density of the residual, and for every N > 0,
converge uniformly in h, l ∈ (−N, N), then the following formula holds.
The asymptotic equivalence of the least square estimates and the Markov estimates is also discussed.
Translated by Wang Ting from the Chinese version of the paper published in Journal of Beijing Normal University (Natural Sciences), 1965, 1: 15–44 相似文献
13.
Mean-value theorems and extensions of the Elliott-Daboussi theorem on additive arithmetic semigroups
Wen-Bin Zhang 《The Ramanujan Journal》2008,15(1):47-75
We present more general forms of the mean-value theorems established before for multiplicative functions on additive arithmetic
semigroups and prove, on the basis of these new theorems, extensions of the Elliott-Daboussi theorem. Let
be an additive arithmetic semigroup with a generating set ℘ of primes p. Assume that the number G(m) of elements a in
with “degree” ∂(a)=m satisfies
with constants q>1, ρ
1<ρ
2<⋅⋅⋅<ρ
r
=ρ, ρ≥1, γ>1+ρ. For the main result, let α,τ,η be positive constants such that α>1,τ
ρ≥1, and τ
α
ρ≥1. Then for a multiplicative function f(a) on
the following two conditions (A) and (B) are equivalent. These are (A) All four series
converge and
and (B) The order τ
ρ mean-value
exists with m
f
≠0 and the limit
exists with M
v
(α)>0.
相似文献
14.
A. V. Ustinov 《Journal of Mathematical Sciences》2006,137(2):4722-4738
Statistical properties of continued fractions for numbers a/b, where a and b lie in the sector a, b ≥ 1, a2 + b2 ≤ R2, are studied. The main result is an asymptotic formula with two meaning terms for the quantity
where sx(a/b) = |{j ε {1, …, s}: [0; tj, …, ts] ≤ x}| is the Gaussian statistic for the fraction a/b = [t0; t1, …, ts]. Bibliography: 12 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 186–211. 相似文献
15.
Abstract
Let Λ = {λ
k
} be an infinite increasing sequence of positive integers with λ
k
→∞. Let X = {X(t), t ∈? R
N
} be a multi-parameter fractional Brownian motion of index α(0 < α < 1) in R
d
. Subject to certain hypotheses, we prove that if N < αd, then there exist positive finite constants K
1 and K
2 such that, with unit probability,
if and only if there exists γ > 0 such that
where ϕ(s) = s
N/α
(log log 1/s)
N/(2α), ϕ-p
Λ(E) is the Packing-type measure of E,X([0, 1])
N
is the image and GrX([0, 1]
N
) = {(t,X(t)); ∈? [0, 1]
N
} is the graph of X, respectively. We also establish liminf type laws of the iterated logarithm for the sojourn measure of X.
Supported by the National Natural Science Foundation of China (No.10471148), Sci-tech Innovation Item for Excellent Young
and Middle-Aged University Teachers and Major Item of Educational Department of Hubei (No.2003A005) 相似文献
16.
Song LI Ruei Fang HU Xiang Qing WANG 《数学学报(英文版)》2006,22(1):51-60
The purpose of this paper is to investigate the solutions of refinement equations of the form ψ（x）∑α∈Z α（α）ψ（Mx-α）,x∈R, where the vector of functions ψ = （ψ1,..., ψr）^T is in （Lp（R^n））^r, 0 〈 p≤∞, α（α）, α ∈ Z^n, is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that limn→∞M^-n=0, In this article, we characterize the existence of an Lp=solution of the refinement equation for 0〈 p ≤∞, Our characterizations are based on the p-norm joint spectral radius. 相似文献
17.
We consider two-phase metrics of the form ϕ(x, ξ) ≔
, where α,β are fixed positive constants and B
α, B
β are disjoint Borel sets whose union is ℝN, and prove that they are dense in the class of symmetric Finsler metrics ϕ satisfying
. Then we study the closure
of the class
of two-phase periodic metrics with prescribed volume fraction θ of the phase α. We give upper and lower bounds for the class
and localize the problem, generalizing the bounds to the non-periodic setting. Finally, we apply our results to study the
closure, in terms of Γ-convergence, of two-phase gradient-constraints in composites of the type f(x, ∇ u) ≤ C(x), with C(x) ∈ {α, β } for almost every x. 相似文献
18.
We study the properties of the polynomial operator pencil
, where
is ak-dimensional Hilbert space, and prove that the mixed discriminants {d
j
}
j=0
nk
, defined as the coefficients of the polynomial
, are completely determined by the joint spectrum of the family {M
i
}
i=0
n
. A generalization of Gershgorin's well-known theorem on the position of the eigenvalues of a matrix to the case of a polynomial
matrix pencil is obtained.
Translated by V. E. Nazaikinskii
Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 3–9, July, 1997. 相似文献
19.
Let M be a compact manifold of dimension n ≥ 2 and 1 < p < n. For a family of functions F
α
defined on TM, which are p-homogeneous, positive, and convex on each fiber, of Riemannian metrics g
α
and of coefficients a
α
on M, we discuss the compactness problem of minimal energy type solutions of the equation
This question is directly connected to the study of the first best constant associated with the Riemannian F
α
-Sobolev inequality
Precisely, we need to know the dependence of under F
α
and g
α
. For that, we obtain its value as the supremum on M of best constants associated with certain homogeneous Sobolev inequalities on each tangent space and show that is attained on M. We then establish the continuous dependence of in relation to F
α
and g
α
. The tools used here are based on convex analysis, blow-up, and variational approach.
相似文献
20.
By using different convex functionals to compute fixed point index, the existence of positive solutions for a class of second-order
two-point boundary value problem
is obtained under some conditions of growth, where α, β, γ, δ ≥ 0, ρ = αγ + γβ + δα > 0, and h(t) is allowed to be singular at t = 0 and t = 1.
Supported by the National Natural Science Foundation of China(10771031,10671167). 相似文献