共查询到20条相似文献,搜索用时 31 毫秒
1.
吴正昌 《高校应用数学学报(英文版)》2001,16(2)
§ 1 IntroductionThe nontrivial solution of the following refinement equation = αa(α)( 2 .-α) ( 1 )is called refinable.The sequence a is called the refinementmask.When the mask a is finite-ly supported on Zand αa(α) =2 ,itis well known thatthe refinementequation( 1 ) hasa unique compactly supported distribution solution satisfies^( 0 ) =1 ,where^denotes theFourier transform of.This solution is called the normalized solution of( 1 ) .In order to study the solution of the equ… 相似文献
2.
Rong-Qing Jia 《Advances in Computational Mathematics》1995,3(4):309-341
Subdivision schemes play an important role in computer graphics and wavelet analysis. In this paper we are mainly concerned
with convergence of subdivision schemes inL
p
spaces (1≤p≤∞). We characterize theL
p
-convergence of a subdivision scheme in terms of thep-norm joint spectral radius of two matrices associated with the corresponding mask. We also discuss various properties of
the limit function of a subdivision scheme, such as stability, linear independence, and smoothness. 相似文献
3.
We consider the existence of distributional (or L
2
) solutions of the matrix refinement equation
where P is an r×r matrix with trigonometric polynomial entries.
One of the main results of this paper is that the above matrix refinement equation has a compactly supported distributional
solution if and only if the matrix P
(0) has an eigenvalue of the form 2
n
, . A characterization of the existence of L
2
-solutions of the above matrix refinement equation in terms of the mask is also given.
A concept of L
2
-weak stability of a (finite) sequence of function vectors is introduced. In the case when the function vectors are solutions
of a matrix refinement equation, we characterize this weak stability in terms of the mask.
August 1, 1996. Date revised: July 28, 1997. Date accepted: August 12, 1997. 相似文献
4.
Yang Wang 《Journal of Approximation Theory》2001,113(2):207
We consider the two-scale refinement equation f(x)=∑Nn=0 cnf(2x−n) with ∑n c2n=∑n c2n+1=1 where c0, cN≠0 and the corresponding subdivision scheme. We study the convergence of the subdivision scheme and the cascade algorithm when all cn0. It has long been conjectured that under such an assumption the subdivision algorithm converge, and the cascade algorithm converge uniformly to a continuous function, if and only if only if 0<c0, cN<1 and the greatest common divisor of S={n: cn>0} is 1. We prove the conjecture for a large class of refinement equations. 相似文献
5.
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
6.
In this paper, we propose a robust semi-explicit difference scheme for solving the Kuramoto–Tsuzuki equation with homogeneous boundary conditions. Because the prior estimate in L∞-norm of the numerical solutions is very hard to obtain directly, the proofs of convergence and stability are difficult for the difference scheme. In this paper, we first prove the second-order convergence in L2-norm of the difference scheme by an induction argument, then obtain the estimate in L∞-norm of the numerical solutions. Furthermore, based on the estimate in L∞-norm, we prove that the scheme is also convergent with second order in L∞-norm. Numerical examples verify the correction of the theoretical analysis. 相似文献
7.
Veli Shakhmurov Muhammet Kurulay 《Numerical Methods for Partial Differential Equations》2023,39(2):923-931
The Lp-coercive properties of a nonlocal fractional elliptic equation is studied. Particularly, it is proved that the fractional elliptic operator generated by this equation is sectorial in Lp space and also is a generator of an analytic semigroup. Moreover, by using the Lp-separability properties of the given elliptic operator the maximal regularity of the corresponding nonlocal fractional parabolic equation is established. 相似文献
8.
Regularity of multiwavelets 总被引:7,自引:0,他引:7
The motivation for this paper is an interesting observation made by Plonka concerning the factorization of the matrix symbol associated with the refinement equation for B-splines with equally spaced multiple knots at integers and subsequent developments which relate this factorization to regularity of refinable vector fields over the real line. Our intention is to contribute to this train of ideas which is partially driven by the importance of refinable vector fields in the construction of multiwavelets. The use of subdivision methods will allow us to consider the problem almost entirely in the spatial domain and leads to exact characterizations of differentiability and Hölder regularity in arbitrary L p spaces. We first study the close relationship between vector subdivision schemes and a generalized notion of scalar subdivision schemes based on bi-infinite matrices with certain periodicity properties. For the latter type of subdivision scheme we will derive criteria for convergence and Hölder regularity of the limit function, which mainly depend on the spectral radius of a bi-infinite matrix induced by the subdivision operator, and we will show that differentiability of the limit functions can be characterized by factorization properties of the subdivision operator. By switching back to vector subdivision we will transfer these results to refinable vectors fields and obtain characterizations of regularity by factorization and spectral radius properties of the symbol associated to the refinable vector field. Finally, we point out how multiwavelets can be generated from orthonormal refinable bi-infinite vector fields. 相似文献
9.
YING Long'an 《中国科学A辑(英文版)》2000,43(9):945-957
A second order explicit finite element scheme is given for the numerical computation to multi-dimensional scalar conservation
laws.L
p
convergence to entropy solutions is proved under some usual conditions. For two-dimensional problems, uniform mesh, and sufficiently
smooth solutions a second order error estimate inL
2 is proved under a stronger condition, Δt≤Ch
2/4 相似文献
10.
Song LI Guo Mao WANG Zhi Song LIU 《数学学报(英文版)》2005,21(6):1475-1486
The purpose of this paper is to investigate the mean size formula of wavelet packets in Lp for 0 〈 p ≤ ∞. We generalize a mean size formula of wavelet packets given in terms of the p-norm joint spectral radius and we also give some asymptotic formulas for the Lp-norm or quasi-norm on the subdivision trees. All results will be given in the general setting, 相似文献
11.
Two-direction refinable functions and two-direction wavelets with high approximation order and regularity 总被引:2,自引:0,他引:2
Shou-zhi YANG~ You-fa LI Department of Mathematics Shantou University Shantou China 《中国科学A辑(英文版)》2007,50(12):1687-1704
The concept of two-direction refinable functions and two-direction wavelets is introduced. We investigate the existence of distributional(or L~2-stable) solutions of the two-direction refinement equation: (?)(x)=(?)p_k~ (?)(mx-k) (?)p_k~-(?)(k-mx), where m≥2 is an integer.Based on the positive mask {p_k~ } and negative mask {p_k~-},the conditions that guarantee the above equation has compactly distributional solutions or L~2-stable solutions are established.Furthermore,the condition that the L~2-stable solution of the above equation can generate a two-direction MRA is given.The support interval of (?)(x) is discussed amply.The definition of orthogonal two-direction refinable function and orthogonal two-direction wavelets is presented,and the orthogonality criteria for two-direction refinable functions are established.An algorithm for construct- ing orthogonal two-direction refinable functions and their two-direction wavelets is presented.Another construction algorithm for two-direction L~2-refinable functions,which have nonnegative symbol masks and possess high approximation order and regularity,is presented.Finally,two construction examples are given. 相似文献
12.
J. Fontbona 《Probability Theory and Related Fields》2006,136(1):102-156
We develop a probabilistic interpretation of local mild solutions of the three dimensional Navier-Stokes equation in the Lp spaces, when the initial vorticity field is integrable. This is done by associating a generalized nonlinear diffusion of
the McKean-Vlasov type with the solution of the corresponding vortex equation. We then construct trajectorial (chaotic) stochastic
particle approximations of this nonlinear process. These results provide the first complete proof of convergence of a stochastic
vortex method for the Navier-Stokes equation in three dimensions, and rectify the algorithm conjectured by Esposito and Pulvirenti
in 1989. Our techniques rely on a fine regularity study of the vortex equation in the supercritical Lp spaces, and on an extension of the classic McKean-Vlasov model, which incorporates the derivative of the stochastic flow
of the nonlinear process to explain the vortex stretching phenomenon proper to dimension three.
Supported by Fondecyt Project 1040689 and Nucleus Millennium Information and Randomness ICM P01-005. 相似文献
13.
Refinable functions with exponential decay arise from applications such as the Butterworth filters in signal processing. Refinable
functions with exponential decay also play an important role in the study of Riesz bases of wavelets generated from multiresolution
analysis. A fundamental problem is whether the standard solution of a refinement equation with an exponentially decaying mask
has exponential decay. We investigate this fundamental problem by considering cascade algorithms in weighted L
p
spaces (1≤p≤∞). We give some sufficient conditions for the cascade algorithm associated with an exponentially decaying mask to converge
in weighted L
p
spaces. Consequently, we prove that the refinable functions associated with the Butterworth filters are continuous functions
with exponential decay. By analyzing spectral properties of the transition operator associated with an exponentially decaying
mask, we find a characterization for the corresponding refinable function to lie in weighted L
2 spaces. The general theory is applied to an interesting example of bivariate refinable functions with exponential decay,
which can be viewed as an extension of the Butterworth filters. 相似文献
14.
《Journal of Computational and Applied Mathematics》1998,98(2):289-304
In this paper, we have proved the second-order convergence in L∞ norm of the Tsertsvadze's difference scheme for the Kuramoto-Tsuzuki equation. The existence, uniqueness and iterative algorithm are also discussed in detail. Furthermore, a L∞ second order convergent linearized difference scheme is given for inhomogeneous equation. All results are obtained without any restrictions on the meshsizes. At last a numerical example is presented. 相似文献
15.
In this paper we study blowup of radially symmetric solutions of the nonlinear heat equation ut = Δu + |u|p?1u either on ?N or on a finite ball under the Dirichlet boundary conditions. We assume that the exponent p is supercritical in the Sobolev sense, that is, We prove that if ps < p < p*, then blowup is always of type I, where p* is a certain (explicitly given) positive number. More precisely, the rate of blowup in the L∞ norm is always the same as that for the corresponding ODE dv/dt = |v|p?1v. Because it is known that “type II” blowup (or, equivalently, “fast blowup”) can occur if p > p*, the above range of exponent p is optimal. We will also derive various fundamental estimates for blowup that hold for any p > ps and regardless of type of blowup. Among other things we classify local profiles of type I and type II blowups in the rescaled coordinates. We then establish useful estimates for the so‐called incomplete blowup, which reveal that incomplete blowup solutions belong to nice function spaces even after the blowup time. © 2004 Wiley Periodicals, Inc. 相似文献
16.
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
17.
In this paper, we discuss the existence and uniqueness of solutions of the constrained variational problem with respect to the fractional Kirchhoff equation. For the exponent p<p*(s,N), a complete classification with respect to p for the existence of solutions of the fractional Kirchhoff functional on the L2-normalized manifold was given. Furthermore, all these solutions are unique up to translations, and our methods depend only on some simple energy estimates. 相似文献
18.
Kenneth L. Wiggins 《Journal of Approximation Theory》1979,26(4):329-339
In this paper best Lp approximate solutions are shown to exist for a wide class of integrodifferential equations. Using approximation theory techniques, a local existence theorem for solutions is established, and the convergence of the best approximate solutions to a solution is shown. 相似文献
19.
Starting with an initial vector λ = (λ(κ))κ∈z ∈ ep(Z), the subdivision scheme generates asequence (Snaλ)∞n=1 of vectors by the subdivision operator Saλ(κ) = ∑λ(j)a(k - 2j), k ∈ Z. j∈zSubdivision schemes play an important role in computer graphics and wavelet analysis. It is very interesting tounderstand under what conditions the sequence (Snaλ)∞n=1 converges to an Lp-function in an appropriate sense.This problem has been studied extensively. In this paper we show that the subdivision scheme converges forany initial vector in ep(Z) provided that it does for one nonzero vector in that space. Moreover, if the integertranslates of the refinable function are stable, the smoothness of the limit function corresponding to the vectorλ is also independent of λ. 相似文献
20.
Ying Guang Shi 《Constructive Approximation》1998,14(2):187-194
A Turán-type inequality for L
p
extremal polynomials is given and mean convergence of Lagrange interpolation based on the zeros of L
p
extremal polynomials is investigated.
November 8, 1994. Date revised: January 23, 1997. 相似文献