Multiwavelet Frames from Refinable Function Vectors

Starting from any two compactly supported d-refinable function vectors in (L ₂(R)) ^r with multiplicity r and dilation factor d, we show that it is always possible to construct 2rd wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L ₂(R) and they achieve the best possible orders of vanishing moments. When all the components of the two real-valued d-refinable function vectors are either symmetric or antisymmetric with their symmetry centers differing by half integers, such 2rd wavelet functions, which generate a pair of dual d-wavelet frames, can be real-valued and be either symmetric or antisymmetric with the same symmetry center. Wavelet frames from any d-refinable function vector are also considered. This paper generalizes the work in [5,12,13] on constructing dual wavelet frames from scalar refinable functions to the multiwavelet case. Examples are provided to illustrate the construction in this paper.     

Pairs of Dual Wavelet Frames from Any Two Refinable Functions

Starting from any two compactly supported refinable functions in L₂(R) with dilation factor d,we show that it is always possible to construct 2d wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L₂(R). Moreover, the number of vanishing moments of each of these wavelet frames is equal to the approximation order of the dual MRA; this is the highest possible. In particular, when we consider symmetric refinable functions, the constructed dual wavelets are also symmetric or antisymmetric. As a consequence, for any compactly supported refinable function in L₂(R), it is possible to construct, explicitly and easily, wavelets that are finite linear combinations of translates (d · – k), and that generate a wavelet frame with an arbitrarily preassigned number of vanishing moments.We illustrate the general theory by examples of such pairs of dual wavelet frames derived from B-spline functions.     

On the periodicity of trigonometric functions generalized to quotient rings of R[x]

We apply a method of Euler to algebraic extensions of sets of numbers with compound additive inverse which can be seen as quotient rings of R[x]. This allows us to evaluate a generalization of Riemann’s zeta function in terms of the period of a function which generalizes the function sin z. It follows that the functions generalizing the trigonometric functions on these sets of numbers are not periodic.     

An improved univariate global optimization algorithm with improved linear lower bounding functions

Recently linear lower bounding functions (LLBF's) were proposed and used to find -global minima. Basically an LLBF over an interval is a linear function which lies below a given function over the interval and matches the function value at one end point. By comparing it with the best function value found, it can be used to eliminate subregions which do not contain -global minima. To develop a more efficient LLBF algorithm, two important issues need to be addressed: how to construct a better LLBF and how to use it efficiently. In this paper, an improved LLBF for factorable functions overn-dimensional boxes is derived, in the sense that the new LLBF is always better than those in [3] for continuously differentiable functions. Exploration of the properties of the LLBF enables us to develop a new LLBF-based univariate global optimization algorithm, which is again better than those in [3]. Numerical results on some standard test functions indicate the high potential of our algorithm.This work was supported in part by VLSI Technology Inc. and Tyecin Systems Inc. through the University of California MICRO proram with grant number 92-024.     

Weighted integral and integro-differential inequalities

A general weighted integral inequality for two continuous functions on an interval [a,b] is presented. The equality conditions are given. This result implies the new inequalities for the incomplete beta and gamma functions as well as the related estimates for the confluent hypergeometric function, error function, and Dawson's integral. Also it implies various weighted integro-differential inequalities, those of the Opial type included, and some inequalities which involve the Erdélyi–Kober and Riemann–Liouville fractional integrals.     

Universal optimality of digital nets and lattice designs

This article considers universal optimality of digital nets and lattice designs in a regression model. Based on the equivalence theorem for matrix means and majorization theory,the necessary and sufficient conditions for lattice designs being φp-and universally optimal in trigonometric function and Chebyshev polynomial regression models are obtained. It is shown that digital nets are universally optimal for both complete and incomplete Walsh function regression models under some specified conditions,and are...     

Ridge Functions and Orthonormal Ridgelets

Orthonormal ridgelets provide an orthonormal basis for L²(R²) built from special angularly-integrated ridge functions. In this paper we explore the relationship between orthonormal ridgelets and true ridge functions r(x₁ cos θ+x₂ sin θ). We derive a formula for the ridgelet coefficients of a ridge function in terms of the 1-D wavelet coefficients of the ridge profile r(t). The formula shows that the ridgelet coefficients of a ridge function are heavily concentrated in ridge parameter space near the underlying scale, direction, and location of the ridge function. It also shows that the rearranged weighted ridgelet coefficients of a ridge function decay at essentially the same rate as the rearranged weighted 1-D wavelet coefficients of the 1-D ridge profile r(t). In short, the full ridgelet expansion of a ridge function is in a certain sense equally as sparse as the 1-D wavelet expansion of the ridge profile. It follows that partial ridgelet expansions can give good approximations to objects which are countable superpositions of well-behaved ridge functions. We study the nonlinear approximation operator which “kills” coefficients below certain thresholds (depending on angular- and ridge-scale); we show that for approximating objects which are countable superpositions of ridge functions with 1-D ridge profiles in the Besov space B^1/p_p, p(R), 0<p<1, the thresholded ridgelet approximation achieves optimal rates of N-term approximation. This implies that appropriate thresholding in the ridgelet basis is equally as good, for certain purposes, as an ideally-adapted N-term nonlinear ridge approximation, based on perfect choice of N-directions.     

Dyadic Bivariate Wavelet Multipliers in L^2（R@2）

The single 2 dilation wavelet multipliers in one-dimensional case and single A-dilation (where A is any expansive matrix with integer entries and |detA| = 2) wavelet multipliers in twodimensional case were completely characterized by Wutam Consortium (1998) and Li Z., et al. (2010). But there exist no results on multivariate wavelet multipliers corresponding to integer expansive dilation matrix with the absolute value of determinant not 2 in L ²(ℝ²). In this paper, we choose $2I_2 = \left( {{*{20}c} 2 & 0 \\ 0 & 2 \\ } \right)$2I_2 = \left( {\begin{array}{*{20}c} 2 & 0 \\ 0 & 2 \\ \end{array} } \right) as the dilation matrix and consider the 2I ₂-dilation multivariate wavelet Φ = {ψ ₁, ψ ₂, ψ ₃}(which is called a dyadic bivariate wavelet) multipliers. Here we call a measurable function family f = {f ₁, f ₂, f ₃} a dyadic bivariate wavelet multiplier if Y₁ = { F ^{- 1} ( f₁ [^(y₁ )] ),F ^{- 1} ( f₂ [^(y₂ )] ),F ^{- 1} ( f₃ [^(y₃ )] ) }\Psi _1 = \left\{ {\mathcal{F}^{ - 1} \left( {f_1 \widehat{\psi _1 }} \right),\mathcal{F}^{ - 1} \left( {f_2 \widehat{\psi _2 }} \right),\mathcal{F}^{ - 1} \left( {f_3 \widehat{\psi _3 }} \right)} \right\} is a dyadic bivariate wavelet for any dyadic bivariate wavelet Φ = {ψ ₁, ψ ₂, ψ ₃}, where [^(f)]\hat f and F ⁻¹ denote the Fourier transform and the inverse transform of function f respectively. We study dyadic bivariate wavelet multipliers, and give some conditions for dyadic bivariate wavelet multipliers. We also give concrete forms of linear phases of dyadic MRA bivariate wavelets.     

Basic Theory in Construction of Boolean Functions with Maximum Possible Annihilator Immunity

So far there is no systematic attempt to construct Boolean functions with maximum annihilator immunity. In this paper we present a construction keeping in mind the basic theory of annihilator immunity. This construction provides functions with the maximum possible annihilator immunity and the weight, nonlinearity and algebraic degree of the functions can be properly calculated under certain cases. The basic construction is that of symmetric Boolean functions and applying linear transformation on the input variables of these functions, one can get a large class of non-symmetric functions too. Moreover, we also study several other modifications on the basic symmetric functions to identify interesting non-symmetric functions with maximum annihilator immunity. In the process we also present an algorithm to compute the Walsh spectra of a symmetric Boolean function with O(n²) time and O(n) space complexity. We use the term “Annihilator Immunity” instead of “Algebraic Immunity” referred in the recent papers [3–5, 9, 18, 19]. Please see Remark 1 for the details of this notational change     

A Criterion of Normality Concerning Holomorphic Functions Whose Derivative Omits a Function

The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of holomorphic functions on a domain D ⊂ ℂ, all of whose zeros have multiplicity at least k, where k ≥ 2 is an integer. And let h(z) ≢ 0 be a holomorphic function on D. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ |f ^(k)(z)| < |h(z)|; (b) f ^(k)(z) ≠ h(z). Then F is normal on D.     

On cardinal wavelets with compact support and their duals

Some people try to construct an orthonormal wavelet such that the corresponding scaling function φ(t) has the cardinal property,i.e. ϕ(n)= σn0, since such wavelets have many good applications. Unfortunately it is impossible to do so, except for a trivial case^[1]. In this work, a family of non-orthogonal cardinal wavelets with compact support is constructed and their duals are investigated. This work is supported by the project of new stars of Beijing     

Property A in an Infinite-Dimensional Space. I

   Abstract. We prove that an infinite-dimensional space of piecewise polynomial functions of degree at most n-1 with infinitely many simple knots, n ≥ 2 , satisfies Property A. Apart from its independent interest, this result allows us to solve an open classical problem (n ≥ 3 ) in theory of best approximation: the uniqueness of best L ₁ -approximation by n -convex functions to an integrable, continuous function defined on a bounded interval. In this first part of the paper we prove the case n=2 and give key results in order to complete the general proof in the second part.     

Transferring strong boundedness among Laguerre orthogonal systems

Given the family of Laguerre polynomials, it is known that several orthonormal systems of Laguerre functions can be considered. In this paper we prove that an exhaustive knowledge of the boundedness in weighted L ^p of the heat and Poisson semigroups, Riesz transforms and g-functions associated to a particular Laguerre orthonormal system of functions, implies a complete knowledge of the boundedness of the corresponding operators on the other Laguerre orthonormal system of functions. As a byproduct, new weighted L ^p boundedness are obtained. The method also allows us to get new weighted estimates for operators related with Laguerre polynomials. Carlos Segovia passed away on April 3, 2007.     

Opérateurs d'extension linéaires explicites dans des intersections de classes ultradifférentiables

B. S. Mityagin proved that the Chebyshev polynomials form a Schauder basis of the space of C ^∞ functions on the interval [–1,1]. Whereof he deduced an explicit continuous linear extension operator. These results were extended, by A. Goncharov, to compact sets without Markov's property. On the reverse, M. Tidten gave examples of compact sets for which there is no continuous linear extension operator. In this paper, we generalize these works to the intersections of ultradifferentiable classes of functions built on the model of the non quasianalytic intersection of Gevrey classes. We get, among other things, a Whitney linear extension theorem for ultradifferentiable jets of Beurling type.     

Quadrature Formulas Based on the Scaling Function

The scaling function corresponding to the Daubechies wavelet with two vanishing moments is used to derive new quadrature formulas. This scaling function has the smallest support among all orthonormal scaling functions with the properties M ₂ = M ₁ ² and M ₀ = 1. So, in this sense, its choice is optimal. Numerical examples are given.This work was partially supported by DFG grant GR 1777/2, by the Grant No 201/01/1200 of the CSF, by the grant MSMT 113200007 and by the grant IGS 116/5130/1 of FP TUL.     

Wavelets on S3 and SO(3)—Their construction,relation to each other and Radon transform of wavelets on SO(3)

The paper at hand is concerned with creating a flexible wavelet theory on the three sphere S³ and the rotation group SO(3). The theory of zonal functions and reproducing kernels will be used to develop conditions for an admissible wavelet. After explaining some preliminaries on group actions and some basics on approximation theory, we will prove reconstruction formulas of linear and bilinear wavelet transformed L²‐functions on S³. Moreover, specific examples will be constructed and visualized. Second, we deal with the construction of wavelets on the rotation group SO(3). It will be shown that the Radon transform of a wavelet packet on SO(3) gives a wavelet packet on S² for every fixed detection direction. Copyright © 2010 John Wiley & Sons, Ltd.     

Truncation Error on Wavelet Sampling Expansions

We estimate the truncation error of sampling expansions on translationinvariant spaces, generated by integer translations of a single functionand on wavelet subspaces of L ₂(R). As a byproduct of themain result, we get the classical Jagerman's bound for Shannon's samplingexpansions. We also examine this error on certain wavelet sampling expansions.     

Ratio asymptotics for orthogonal rational functions on an interval

Let {α₁,α₂,…} be a sequence of real numbers outside the interval [−1,1] and μ a positive bounded Borel measure on this interval satisfying the Erd s–Turán condition μ′>0 a.e., where μ′ is the Radon–Nikodym derivative of the measure μ with respect to the Lebesgue measure. We introduce rational functions _n(x) with poles {α₁,…,α_n} orthogonal on [−1,1] and establish some ratio asymptotics for these orthogonal rational functions, i.e. we discuss the convergence of _n+1(x)/_n(x) as n tends to infinity under certain assumptions on the location of the poles. From this we derive asymptotic formulas for the recurrence coefficients in the three-term recurrence relation satisfied by the orthonormal functions.     

Walsh-Type Wavelet Packet Expansions

We consider a family of basic nonstationary wavelet packets generated using the Haar filters except for a finite number of scales where we allow the use of arbitrary filters. Such a system, which we call a system of Walsh-type wavelet packets, can be considered as a smooth generalization of the Walsh functions. We show that the basic Walsh-type wavelet packets share a number of metric properties with the Walsh system. We prove that the system constitutes a Schauder basis for L^p( ), 1<p<∞, and we construct an explicit function in L¹( ) for which the expansion fails. Then we prove that expansions of L^p( )-functions, 1<p<∞, in the Walsh-type wavelet packets converge pointwise a.e. Finally, we prove that the analogous results are true for periodic Walsh-type wavelet packets in L^p[0,1).     

N–Term Approximation in Anisotropic Function Spaces

In L₂(0, 1)²) infinitely many different biorthogonal wavelet bases may be introduced by taking tensor products of one–dimensional biorthogonal wavelet bases on the interval (0, 1). Most well–known are the standard tensor product bases and the hyperbolic bases. In [23, 24] further biorthogonal wavelet bases are introduced, which provide wavelet characterizations for functions in anisotropic Besov spaces. Here we address the following question: Which of those biorthogonal tensor product wavelet bases is the most appropriate one for approximating nonlinearly functions from anisotropic Besov spaces? It turns out, that the hyperbolic bases lead to nonlinear algorithms which converge as fast as the corresponding schemes with respect to specific anisotropy adapted bases.