On construction of multivariate wavelets with vanishing moments

Wavelets with matrix dilation are studied. An explicit formula for masks providing vanishing moments is found. The class of interpolatory masks providing vanishing moments is also described. For an interpolatory mask, formulas for a dual mask which also provides vanishing moments of the same order and for wavelet masks are given explicitly. An example of construction of symmetric and antisymmetric wavelets for a concrete matrix dilation is presented.     

On harmonic renewal measures

Summary Let be a probability and the corresponding harmonic renewal measure. Complementing earlier results where is concentrated on a halfline we investigate the behaviour ofv _h ([x, x + 1]) and the harmonic renewal functionG(x) =v _h((–,x])asx ifm ₁=x(dx)>0. We also consider the casem ₁=0.     

Construction of orthonormal multi-wavelets with additional vanishing moments

An iterative scheme for constructing compactly supported orthonormal (o.n.) multi-wavelets with vanishing moments of arbitrarily high order is established. Precisely, let φ=[φ₁,. . .,φ_r]^⊤ be an r-dimensional o.n. scaling function vector with polynomial preservation of order (p.p.o.) m, and ψ=[ψ₁,. . .,ψ_r]^⊤ an o.n. multi-wavelet corresponding to φ, with two-scale symbols P and Q, respectively. Then a new (r+1)-dimensional o.n. scaling function vector φ^♯:=[φ^⊤,φ_r+1]^⊤ and some corresponding o.n. multi-wavelet ψ^♯ are constructed in such a way that φ^♯ has p.p.o.=n>m and their two-scale symbols P^♯ and Q^♯ are lower and upper triangular block matrices, respectively, without increasing the size of the supports. For instance, for r=1, if we consider the mth order Daubechies o.n. scaling function φ_m^D, then φ^♯:=[φ_m^D,φ₂]^⊤ is a scaling function vector with p.p.o. >m. As another example, for r=2, if we use the symmetric o.n. scaling function vector φ in our earlier work, then we obtain a new pair of scaling function vector φ^♯=[φ^⊤,φ₃]^⊤ and multi-wavelet ψ^♯ that not only increase the order of vanishing moments but also preserve symmetry. Dedicated to Charles A. Micchelli in Honor of His Sixtieth Birthday Mathematics subject classifications (2000) 42C15, 42C40. Charles K. Chui: Supported in part by NSF grants CCR-9988289 and CCR-0098331 and Army Research Office under grant DAAD 19-00-1-0512. Jian-ao Lian: Supported in part by Army Research Office under grant DAAD 19-01-1-0739.     

On the moments of complex measures

Ohne Zusammenfassung     

Construction of multivariate biorthogonal wavelets with arbitrary vanishing moments

We present a concrete method to build discrete biorthogonal systems such that the wavelet filters have any number of vanishing moments. Several algorithms are proposed to construct multivariate biorthogonal wavelets with any general dilation matrix and arbitrary order of vanishing moments. Examples are provided to illustrate the general theory and the advantages of the algorithms. This revised version was published online in June 2006 with corrections to the Cover Date.     

具有一定消失矩的优化线性相位滤波器

利用消失矩特性和编码误差最小,给出了一类有限长线性相位的双正交小波滤波器组BNVF的构造方法,BNVF的综合低通和分解高通的系数为二进分数,将其用于图像的分解与重构时,可避免一些乘法运算．三组BNVF用于图像变换编码时,其压缩性能不亚于Cohen,Daubechies等构造的9／7步滤波器组CDF-9／7,且计算复杂度更低．     

Compactly supported tight and sibling frames with maximum vanishing moments

The notion of vanishing-moment recovery (VMR) functions is introduced in this paper for the construction of compactly supported tight frames with two generators having the maximum order of vanishing moments as determined by the given refinable function, such as the mth order cardinal B-spline N_m. Tight frames are also extended to “sibling frames” to allow additional properties, such as symmetry (or antisymmetry), minimum support, “shift-invariance,” and inter-orthogonality. For N_m, it turns out that symmetry can be achieved for even m and antisymmetry for odd m, that minimum support and shift-invariance can be attained by considering the frame generators with two-scale symbols 2^−m(1−z)^m and 2^−mz(1−z)^m, and that inter-orthogonality is always achievable, but sometimes at the sacrifice of symmetry. The results in this paper are valid for all compactly supported refinable functions that are reasonably smooth, such as piecewise Lipα for some α>0, as long as the corresponding two-scale Laurent polynomial symbols vanish at z=−1. Furthermore, the methods developed here can be extended to the more general setting, such as arbitrary integer scaling factors, multi-wavelets, and certainly biframes (i.e., allowing the dual frames to be associated with a different refinable function).     

Moment vanishing properties of harmonic Bergman functions

We show that Poisson integrals belonging to certain weighted harmonic Bergman spaces b_δ^p on the upper half-space must have the moment vanishing properties. As an application, we show that b₀^p, p?1, contains a dense subspace whose members have the horizontal moment vanishing properties. Also, we derive related weighted norm inequalities for Poisson integrals. As a consequence, we obtain a characterization for Poisson integrals of continuous functions with compact support in order to belong to b_δ^p.     

On surfaces with zero vanishing cycles

We show that using an idea from a paper by Van de Ven one may obtain a simple proof of Zak’s classification of smooth projective surfaces with zero vanishing cycles. This method of proof allows one to extend Zak’s theorem to the case of finite characteristic.     

On stable hypersurfaces with vanishing scalar curvature

We will prove that there are no stable complete hypersurfaces of $\mathbb {R}^4$ with zero scalar curvature, polynomial volume growth and such that $\frac{(-K)}{H^3}\ge c>0$ everywhere, for some constant $c>0$ , where K denotes the Gauss-Kronecker curvature and $H$ denotes the mean curvature of the immersion. Our second result is the Bernstein type one there is no entire graphs of $\mathbb {R}^4$ with zero scalar curvature such that $\frac{(-K)}{H^3}\ge c>0$ everywhere. At last, it will be proved that, if there exists a stable hypersurface with zero scalar curvature and $\frac{(-K)}{H^3}\ge c>0$ everywhere, that is, with volume growth larger than polynomial growth of order four, then its tubular neighborhood is not embedded for suitable radius.     

Analytic signals and harmonic measures

We prove that a sufficient and necessary condition for HeiΘ(s)=−ieiΘ(s), where H is Hilbert transformation, Θ is a continuous and strictly increasing function with |Θ(R)|=2π, is that dΘ(s) is a harmonic measure on the line. The counterpart result for the periodic case is also established. The study is motivated by, and has significant impact to time-frequency analysis, especially to aspects of analytic signals inducing instantaneous amplitude and frequency. As a by-product we introduce the theory of Hardy-space-preserving weighted trigonometric series and Fourier transformations induced by harmonic measures in the respective contexts.     

Convexity properties of harmonic measures

It is shown that any convex combination of harmonic measures , where U₁,…,Uk are relatively compact open neighborhoods of a given point x∈Rd, d?2, can be approximated by a sequence of harmonic measures such that each Wn is an open neighborhood of x in U₁∪?∪Uk.This answers a question raised in connection with Jensen measures. Moreover, it implies that, for every Green domain X containing x, the extremal representing measures for x with respect to the convex cone of potentials on X (these measures are obtained by balayage of the Dirac measure at x on Borel subsets of X) are dense in the compact convex set of all representing measures.This is achieved approximating balayage on open sets by balayage on unions of balls which are pairwise disjoint and very small with respect to their mutual distances and then reducing the size of these balls in a suitable manner.These results, which are presented simultaneously for the classical potential theory and for the theory of Riesz potentials, can be sharpened if the complements or the boundaries of the open sets have a capacity doubling property. The methods developed for this purpose (continuous balayage on increasing families of compact sets, approximation using scattered sets with small capacity) finally lead to answers even in a very general potential-theoretic setting covering a wide class of second order partial differential operators (uniformly elliptic or in divergence form, or sums of squares of vector fields satisfying Hörmander's condition, for example, sub-Laplacians on stratified Lie algebras).     

On duality for mathematical programs with vanishing constraints

We make a review of several variants of ergodicity for continuous-time Markov chains on a countable state space. These include strong ergodicity, ergodicity in weighted-norm spaces, exponential and subexponential ergodicity. We also study uniform exponential ergodicity for continuous-time controlled Markov chains, as a tool to deal with average reward and related optimality criteria. A discussion on the corresponding ergodicity properties is made, and an application to a controlled population system is shown.