ScholarBank@NUShttps://scholarbank.nus.edu.sgThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Thu, 30 Mar 2023 02:03:08 GMT2023-03-30T02:03:08Z50131- Compactly Supported Refinable Distributions in Triebel-Lizorkin Spaces and Besov Spaceshttps://scholarbank.nus.edu.sg/handle/10635/103007Title: Compactly Supported Refinable Distributions in Triebel-Lizorkin Spaces and Besov Spaces
Authors: Ma, B.; Sun, Q.
Abstract: The aim of this article is to characterize compactly supported refinable distributions in Triebel-Lizorkin spaces and Besov spaces by projection operators on certain wavelet space and by some operators on a finitely dimensional space.
Fri, 01 Jan 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1030071999-01-01T00:00:00Z
- Compactly Supported Distributional Solutions of Nonstationary Nonhomogeneous Refinement Equationshttps://scholarbank.nus.edu.sg/handle/10635/103006Title: Compactly Supported Distributional Solutions of Nonstationary Nonhomogeneous Refinement Equations
Authors: Sun, Q.Y.
Abstract: Let A be a matrix with the absolute values of all eigenvalues strictly larger than one, and let Zo be a subset of Z such that n ∈ Z0 implies n + 1 ∈ Z0. Denote the space of all compactly supported distributions by D′, and the usual convolution between two compactly supported distributions f and g by f* g. For any bounded sequence Gn and Hn, n ∈ Z0, in D′, define the corresponding nonstationary nonhomogeneous refinement equation Φn = Hn * Φn+1 (A ·) + Gn for all n ∈ Z0, (*) where Φ n,n ∈ Z0, is in a bounded set of D′. The nonstationary nonhomogeneous refinement equation (*) arises in the construction of wavelets on bounded domain, multiwavelets, and of biorthogonal wavelets on nonuniform meshes. In this paper, we study the existence problem of compactly supported distributional solutions Φn,n ∈ Zo, of the equation (*). In fact, we reduce the existence problem to finding a bounded solution F̃n of the linear equations F̃n - SnF̃n+1 = G̃n for all n ∈ Z0, where the matrices Sn and the vectors G̃n, n ∈ Zo, can be constructed explicitly from Hn and Gn respectively. The results above are still new even for stationary nonhomogeneous refinement equations.
Mon, 01 Jan 2001 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1030062001-01-01T00:00:00Z
- Construction of compactly supported M-band waveletshttps://scholarbank.nus.edu.sg/handle/10635/103052Title: Construction of compactly supported M-band wavelets
Authors: Bi, N.; Dai, X.; Sun, Q.
Abstract: In this paper, we consider the asymptotic regularity of Daubechies scaling functions and construct examples of M-band scaling functions which are both orthonormal and cardinal for M ≥ 3. © 1999 Academic Press.
Fri, 01 Jan 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1030521999-01-01T00:00:00Z
- A class of m-dilation scaling functions with regularity growing proportionally to filter support widthhttps://scholarbank.nus.edu.sg/handle/10635/102614Title: A class of m-dilation scaling functions with regularity growing proportionally to filter support width
Authors: Shi, X.; Sun, Q.
Abstract: In this paper, a class of M-dilation scaling functions with regularity growing proportionally to filter support width is constructed. This answers a question proposed by Daubechies on p.338 of her book Ten Lectures on Wavelets (1992). © 1998 American Mathematical Society.
Thu, 01 Jan 1998 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1026141998-01-01T00:00:00Z
- Asymptotic regularity of daubechies' scaling functionshttps://scholarbank.nus.edu.sg/handle/10635/102897Title: Asymptotic regularity of daubechies' scaling functions
Authors: Lau, K.A.-S.; Sun, Q.
Abstract: Let φN, N ≥ 1, be Daubechies' scaling function with symbol (1+e-iξ/2) QN(ξ) and let Sp(φN),0 < p ≤ ∞, be the corresponding Lp Sobolev exponent. In this paper, we make a sharp estimation of Sp(φN), and we prove that there exists a constant C independent of N such that Matrix Equation Presented This answers a question of Cohen and Daubeschies (Rev. Mat. Iberoamericana, 12(1996), 527-591) positively. © 2000 American Mathematical Society.
Sat, 01 Jan 2000 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1028972000-01-01T00:00:00Z
- Sobolev exponent estimate and asymptotic regularity of the M-band Daubechies' scaling functionshttps://scholarbank.nus.edu.sg/handle/10635/104141Title: Sobolev exponent estimate and asymptotic regularity of the M-band Daubechies' scaling functions
Authors: Qiyu, S.
Abstract: In this paper, the direct estimate of the Sobolev exponent of refinable distributions and its application to the asymptotic estimate of the Sobolev exponent of the M-band Daubechies' scaling functions are considered.
Fri, 01 Jan 1999 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1041411999-01-01T00:00:00Z
- Stability of the shifts of global supported distributionshttps://scholarbank.nus.edu.sg/handle/10635/104190Title: Stability of the shifts of global supported distributions
Authors: Sun, Q.
Abstract: For a tempered distribution with ℓ1 decay, we characterize its stable shifts via its Fourier transform and via a shift-invariant space of summable sequences. Also we show that if the tempered distribution with ℓ1 decay has stable shifts, then we can recover all distributions in V∞, the space of all linear combinations of its shifts using bounded sequences, in a stable way using C∞ dual functions with ℓ1 decay at infinity. If, additionally, that tempered distribution is compactly supported, then the above C∞ dual functions can be chosen to have exponential decay at infinity. © 2001 Academic Press.
Sat, 01 Sep 2001 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1041902001-09-01T00:00:00Z
- Compactly supported tight affine frames with integer dilations and maximum vanishing momentshttps://scholarbank.nus.edu.sg/handle/10635/103009Title: Compactly supported tight affine frames with integer dilations and maximum vanishing moments
Authors: Chui, C.K.; He, W.; Stöckler, J.; Sun, Q.
Abstract: When a cardinal B-spline of order greater than 1 is used as the scaling function to generate a multiresolution approximation of L2 = L 2(ℝ) with dilation integer factor M ≥ 2, the standard "matrix extension" approach for constructing compactly supported tight frames has the limitation that at least one of the tight frame generators does not annihilate any polynomial except the constant. The notion of vanishing moment recovery (VMR) was introduced in our earlier work (and independently by Daubechies et al.) for dilation M = 2 to increase the order of vanishing moments. This present paper extends the tight frame results in the above mentioned papers from dilation M = 2 to arbitrary integer M ≥ 2 for any compactly supported M-dilation scaling functions. It is shown, in particular, that M compactly supported tight frame generators suffice, but not M - 1 in general. A complete characterization of the M-dilation polynomial symbol is derived for the existence of M - 1 such frame generators. Linear spline examples are given for M = 3, 4 to demonstrate our constructive approach.
Sat, 01 Feb 2003 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1030092003-02-01T00:00:00Z
- Algorithm for the construction of symmetric and anti-symmetric M-band waveletshttps://scholarbank.nus.edu.sg/handle/10635/104529Title: Algorithm for the construction of symmetric and anti-symmetric M-band wavelets
Authors: Sun, Qiyu
Abstract: In this paper, we give an algorithm to construct semi-orthogonal symmetric and anti-symmetric M-band wavelets. As an application, some semi-orthogonal symmetric and anti-symmetric M-band spline wavelets are constructed explicitly. Also we show that if we want to construct symmetric or anti-symmetric M-band wavelets from a multiresolution, then that multiresolution has a symmetric scaling function.
Sat, 01 Jan 2000 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1045292000-01-01T00:00:00Z
- Convergence of cascade algorithms and smoothness of refinable distributionshttps://scholarbank.nus.edu.sg/handle/10635/103069Title: Convergence of cascade algorithms and smoothness of refinable distributions
Authors: Sun, Q.
Abstract: In this paper, the author at first develops a method to study convergence of the cascade algorithm in a Banach space without stable assumption on the initial (see Theorem 2.1), and then applies the previous result on the convergence to characterizing compactly supported refinable distributions in fractional Sobolev spaces and Hölder continuous spaces (see Theorems 3.1, 3.3, and 3.4). Finally the author applies the above characterization to choosing appropriate initial to guarantee the convergence of the cascade algorithm (see Theorem 4.2).
Wed, 01 Jan 2003 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1030692003-01-01T00:00:00Z
- Eigenvalues of scaling operators and a characterization of B-splineshttps://scholarbank.nus.edu.sg/handle/10635/103192Title: Eigenvalues of scaling operators and a characterization of B-splines
Authors: GAO XIAOJIE; Lee, S.L.; Sun, Q.
Abstract: A finitely supported sequence a that sums to 2 defines a scaling operator Taf = ∑k∈z a(k) f (2·- k) on functions f, a transition operator Sav = ∑k∈z a(k) (2·- k) on sequences v, and a unique compactly supported scaling function φ that satisfies φ = Taφ normalized with φ̂(0) = 1. It is shown that the eigenvalues of Ta on the space of compactly supported square-integrable functions are a subset of the nonzero eigenvalues of the transition operator Sas on the space of finitely supported sequences, and that the two sets of eigenvalues are equal if and only if the corresponding scaling function φ is a uniform B-spline. ©2005 American Mathematical Society.
Sat, 01 Apr 2006 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1031922006-04-01T00:00:00Z
- Tight frame oversampling and its equivalence to shift-invariance of affine frame operatorshttps://scholarbank.nus.edu.sg/handle/10635/104646Title: Tight frame oversampling and its equivalence to shift-invariance of affine frame operators
Authors: Chui, C.K.; Sun, Q.
Abstract: Let Ψ = {ψ1,..., ψL} ⊂ L2 := L2(-∞, ∞) generate a tight affine frame with dilation factor M, where 2 ≤ M ∈ Z, and sampling constant b = 1 (for the zeroth scale level). Then for 1 ≤ N ∈ Z, N × oversampling (or oversampling by N) means replacing the sampling constant 1 by 1/N. The Second Oversampling Theorem asserts that N × oversampling of the given tight affine frame generated by Ψ preserves a tight affine frame, provided that N = N0 is relatively prime to M (i.e., gcd(N0, M) = 1). In this paper, we discuss the preservation of tightness in mN0 × oversampling, where 1 ≤ m|M (i.e., 1 ≤ m ≤ M and gcd(m, M) = m). We also show that tight affine frame preservation in mN0 × oversampling is equivalent to the property of shift-invariance with respect to 1/mN0 of the affine frame operator Q0,N0 defined on the zeroth scale level.
Thu, 01 May 2003 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1046462003-05-01T00:00:00Z
- Local polynomial property and linear independence of refinable distributionshttps://scholarbank.nus.edu.sg/handle/10635/103513Title: Local polynomial property and linear independence of refinable distributions
Authors: Dai, X.; Huang, D.; Sun, Q.
Abstract: In this paper, local polynomial property, global linear independence, and local linear dependence of the convolution of a B-spline and a refinable distribution supported on a Cantor-like set are studied.
Tue, 01 Jan 2002 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/1035132002-01-01T00:00:00Z