Inhomogeneous refinement equations

Equations with two time scales (refinement equations or dilation equations) are central to wavelet theory. Several applications also include an inhomogeneous forcing term F(t). We develop here a part of the existence theory for the inhomogeneous refinement equation

Dual Wavelet Frames and Riesz Bases in Sobolev Spaces

This paper generalizes the mixed extension principle in L ₂(ℝ ^d ) of (Ron and Shen in J. Fourier Anal. Appl. 3:617–637, 1997) to a pair of dual Sobolev spaces H ^s (ℝ ^d ) and H ^−s (ℝ ^d ). In terms of masks for φ,ψ ¹,…,ψ ^L ∈H ^s (ℝ ^d ) and , simple sufficient conditions are given to ensure that (X ^s (φ;ψ ¹,…,ψ ^L ), forms a pair of dual wavelet frames in (H ^s (ℝ ^d ),H ^−s (ℝ ^d )), where

Classification of Refinable Splines in ℝ d

A refinable spline in ℝ ^d is a compactly supported refinable function whose support can be decomposed into simplices such that the function is a polynomial on each simplex. The best-known refinable splines in ℝ ^d are the box splines. Refinable splines play a key role in many applications, such as numerical computation, approximation theory and computer-aided geometric design. Such functions have been classified in one dimension in Dai et al. (Appl. Comput. Harmon. Anal. 22(3), 374–381, 2007), Lawton et al. (Comput. Math. 3, 137–145, 1995). In higher dimensions Sun (J. Approx. Theory 86, 240–252, 1996) characterized those splines when the dilation matrices are of the form A=mI, where m∈ℤ and I is the identity matrix. For more general dilation matrices the problem becomes more complex. In this paper we give a complete classification of refinable splines in ℝ ^d for arbitrary dilation matrices A∈M _d (ℤ).     

Averaging distances in real quasihypermetric Banach spaces of finite dimension

The average distance theorem of Gross implies that for each realN-dimensional Banach space (N≥2) there is a unique positive real numberr(E) with the following property: For each positive integern and for all (not necessarily distinct)x ₁,x ₂, …,x _n inE with ‖x ₁‖=‖x ₂‖=…=‖x _n‖=1, there exists anx inE with ‖x‖=1 such that The main result of this paper shows, thatr(E)≤2−1/N for each realN-dimensional Banach spaceE (N≥2) with the so-called quasihypermetric property (which is equivalent toE isL ¹-embeddable). Moreover, equality holds if and only ifE is isometrically isomorphic to ℝ ^N equipped with the usual 1-norm.     

Some simple Haar-type wavelets in higher dimensions

An orthonormal wavelet system in ℝ^d, d ∈ ℕ, is a countable collection of functions {ψ _j,k ^ℓ }, j ∈ ℤ, k ∈ ℤ^d, ℓ = 1,..., L, of the form that is an orthonormal basis for L² (ℝ^d), where a ∈ GL_d (ℝ) is an expanding matrix. The first such system to be discovered (almost 100 years ago) is the Haar system for which L = d = 1, ψ¹(x) = ψ(x) = κ[0,1/2)^(x) − κ_[l/2,1) ^(x), a = 2. It is a natural problem to extend these systems to higher dimensions. A simple solution is found by taking appropriate products Φ(x₁, x₂, ..., x_d) = φ₁ (x₁)φ₂(x₂) ... φ_d(x_d) of functions of one variable. The obtained wavelet system is not always convenient for applications. It is desirable to find “nonseparable” examples. One encounters certain difficulties, however, when one tries to construct such MRA wavelet systems. For example, if a = ( _1-1 ^{1 1} ) is the quincunx dilation matrix, it is well-known (see, e.g., [5]) that one can construct nonseparable Haar-type scaling functions which are characteristic functions of rather complicated fractal-like compact sets. In this work we shall construct considerably simpler Haar-type wavelets if we use the ideas arising from “composite dilation” wavelets. These were developed in [7] and involve dilations by matrices that are products of the form a^jb, j ∈ ℤ, where a ∈ GL_d(ℝ) has some “expanding” property and b belongs to a group of matrices in GL_d(ℝ) having |det b| = 1.     

Construction of multivariate compactly supported orthonormal wavelets

We propose a constructive method to find compactly supported orthonormal wavelets for any given compactly supported scaling function φ in the multivariate setting. For simplicity, we start with a standard dilation matrix 2I_2×2 in the bivariate setting and show how to construct compactly supported functions ψ₁,. . .,ψ_n with n>3 such that {2^kψ_j(2^kx−ℓ,2^ky−m), k,ℓ,m∈Z, j=1,. . .,n} is an orthonormal basis for L₂(ℝ²). Here, n is dependent on the size of the support of φ. With parallel processes in modern computer, it is possible to use these orthonormal wavelets for applications. Furthermore, the constructive method can be extended to construct compactly supported multi-wavelets for any given compactly supported orthonormal multi-scaling vector. Finally, we mention that the constructions can be generalized to the multivariate setting. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday Mathematics subject classifications (2000) 42C15, 42C30.     

Orthogonality criteria for compactly supported refinable functions and refinable function vectors

A refinable function φ(x):ℝⁿ→ℝ or, more generally, a refinable function vector Φ(x)=[φ₁(x),...,φ_r(x)]^T is an L¹ solution of a system of (vector-valued) refinement equations involving expansion by a dilation matrix A, which is an expanding integer matrix. A refinable function vector is called orthogonal if {φ_j(x−α):α∈ℤⁿ, 1≤j≤r form an orthogonal set of functions in L²(ℝⁿ). Compactly supported orthogonal refinable functions and function vectors can be used to construct orthonormal wavelet and multiwavelet bases of L²(ℝⁿ). In this paper we give a comprehensive set of necessary and sufficient conditions for the orthogonality of compactly supported refinable functions and refinable function vectors.     

A Characterization of Quadric Constant Mean Curvature Hypersurfaces of Spheres

Let be an immersion of a complete n-dimensional oriented manifold. For any v∈ℝ ⁿ⁺², let us denote by ℓ _v :M→ℝ the function given by ℓ _v (x)=〈φ(x),v〉 and by f _v :M→ℝ, the function given by f _v (x)=〈ν(x),v〉, where is a Gauss map. We will prove that if M has constant mean curvature, and, for some v≠0 and some real number λ, we have that ℓ _v =λ f _v , then, φ(M) is either a totally umbilical sphere or a Clifford hypersurface. As an application, we will use this result to prove that the weak stability index of any compact constant mean curvature hypersurface M ⁿ in which is neither totally umbilical nor a Clifford hypersurface and has constant scalar curvature is greater than or equal to 2n+4. A. Brasil Jr. was partially supported by CNPq, Brazil, 306626/2007-1.     

Real Paley-Wiener type theorems for the Dunkl transform on {\mathcal{S}}'(\mathbb{R}^{d})

We prove real Paley-Wiener type theorems for the Dunkl transform ℱ _D on the space of tempered distributions. Let T∈S′(ℝ ^d ) and Δ _κ the Dunkl Laplacian operator. First, we establish that the support of ℱ _D (T) is included in the Euclidean ball , M>0, if and only if for all R>M we have lim  _n→+∞ R ⁻²ⁿ Δ _κ ⁿ T=0 in S′(ℝ ^d ). Second, we prove that the support of ℱ _D (T) is included in ℝ ^d ∖B(0,M), M>0, if and only if for all R<M, we have lim  _n→+∞ R ²ⁿ  ℱ _D ⁻¹(‖y‖⁻²ⁿ ℱ _D (T))=0 in S′(ℝ ^d ). Finally, we study real Paley-Wiener theorems associated with -slowly increasing function.      

Closure Properties of Solutions to Heat Inequalities

We prove that if u ₁,u ₂:(0,∞)×ℝ ^d →(0,∞) are sufficiently well-behaved solutions to certain heat inequalities on ℝ ^d then the function u:(0,∞)×ℝ ^d →(0,∞) given by also satisfies a heat inequality of a similar type provided . On iterating, this result leads to an analogous statement concerning n-fold convolutions. As a corollary, we give a direct heat-flow proof of the sharp n-fold Young convolution inequality and its reverse form. Both authors were supported by EPSRC grant EP/E022340/1.     

Hölder Continuity of Random Processes

For a Young function φ and a Borel probability measure m on a compact metric space (T,d) the minorizing metric is defined by

Traversing a Set of Points with a Minimum Number of Turns

Given a finite set of points S in ℝ ^d , consider visiting the points in S with a polygonal path which makes a minimum number of turns, or equivalently, has the minimum number of segments (links). We call this minimization problem the minimum link spanning path problem. This natural problem has appeared several times in the literature under different variants. The simplest one is that in which the allowed paths are axis-aligned. Let L(S) be the minimum number of links of an axis-aligned path for S, and let G _n ^d be an n×…×n grid in ℤ ^d . Kranakis et al. (Ars Comb. 38:177–192, 1994) showed that L(G _n ²)=2n−1 and and conjectured that, for all d≥3, We prove the conjecture for d=3 by showing the lower bound for L(G _n ³). For d=4, we prove that For general d, we give new estimates on L(G _n ^d ) that are very close to the conjectured value. The new lower bound of improves previous result by Collins and Moret (Inf. Process. Lett. 68:317–319, 1998), while the new upper bound of differs from the conjectured value only in the lower order terms. For arbitrary point sets, we include an exact bound on the minimum number of links needed in an axis-aligned path traversing any planar n-point set. We obtain similar tight estimates (within 1) in any number of dimensions d. For the general problem of traversing an arbitrary set of points in ℝ ^d with an axis-aligned spanning path having a minimum number of links, we present a constant ratio (depending on the dimension d) approximation algorithm. Work by A. Dumitrescu was partially supported by NSF CAREER grant CCF-0444188. Work by F. Hurtado was partially supported by projects MECMTM2006-01267 and Gen. Cat. 2005SGR00692. Work by P. Valtr was partially supported by the project 1M0545 of the Ministry of Education of the Czech Republic.     

Affine Synthesis onto L p when 0<p≤1

The affine synthesis operator is shown to map the coefficient space ℓ ^p (ℤ₊×ℤ ^d ) surjectively onto L ^p (ℝ ^d ), for p∈(0,1]. Here ψ _j,k (x)=|det a _j |^1/p ψ(a _j x−k) for dilation matrices a _j that expand, and the synthesizer ψ∈L ^p (ℝ ^d ) need satisfy only mild restrictions, for example, ψ∈L ¹(ℝ ^d ) with nonzero integral or else with periodization that is real-valued, nontrivial and bounded below. An affine atomic decomposition of L ^p follows immediately:

Trust-region and other regularisations of linear least-squares problems

We consider methods for regularising the least-squares solution of the linear system Ax=b. In particular, we propose iterative methods for solving large problems in which a trust-region bound ‖x‖≤Δ is imposed on the size of the solution, and in which the least value of linear combinations of ‖Ax−b‖₂ ^q and a regularisation term ‖x‖₂ ^p for various p and q=1,2 is sought. In each case, one or more “secular” equations are derived, and fast Newton-like solution procedures are suggested. The resulting algorithms are available as part of the ALAHAD optimization library. This work was partially supported by EPSRC grants EP/E053351/1 and EP/F005369/1.     

Convergence of cascade algorithm for individual initial function and arbitrary refinement masks

The cascade algorithm plays an important role in computer graphics and wavelet analysis. For any initial function φn, a cascade sequence (φn)n∞=1 constructed by the iteration φn=Cnφn-1=1,2.. where Cαis defined by g∈Lp(R) In this paper, we characterize the convergence of a cascade sequence in terms of a sequence of functions and in terms of joint spectral radius. As a consequence, it is proved that any convergent cascade sequence has a convergence rate of geometry, i.e., ||φ 1-φn||Lp(R)=O((？)n)for some (？)∈(0.1i). The condition of sum rules for the mask is not required. Finally, an example is presented to illustrate our theory.     

Extension of Floquet''''s theory to nonlinear quasiperiodic differential equations

In this paper, we consider the following autonomous system of differential equations: x = Ax f(x,θ), θ = ω, where θ∈Rm, ω = (ω1,…,ωm) ∈ Rm, x ∈ Rn, A ∈ Rn×n is a constant matrix and is hyperbolic, f is a C∞ function in both variables and 2π-periodic in each component of the vector e which satisfies f = O(||x||2) as x → 0. We study the normal form of this system and prove that under some proper conditions this system can be transformed to an autonomous system: x = Ax g(x), θ = ω. Additionally, the proof of this paper naturally implies the extension of Chen's theory in the quasi-periodic case.     

A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane

Summary  We consider the numerical treatment of second kind integral equations on the real line of the form

Asymptotics of Recurrence Relation Coefficients,Hankel Determinant Ratios,and Root Products Associated with Laurent Polynomials Orthogonal with Respect to Varying Exponential Weights

Let Λ^ℝ denote the linear space over ℝ spanned by z ^k , k∈ℤ. Define the real inner product 〈⋅,⋅〉 _L :Λ^ℝ×Λ^ℝ→ℝ, , N∈ℕ, where V satisfies: (i) V is real analytic on ℝ∖{0}; (ii) lim  _{| x |→∞}(V(x)/ln (x ²+1))=+∞; and (iii) lim  _{| x |→0}(V(x)/ln (x ⁻²+1))=+∞. Orthogonalisation of the (ordered) base with respect to 〈⋅,⋅〉 _L yields the even degree and odd degree orthonormal Laurent polynomials (OLPs) : φ _2n (z)=∑ _k=−n ⁿ ξ _k ⁽²ⁿ⁾ z ^k , ξ _n ⁽²ⁿ⁾>0, and φ _2n+1(z)=∑ _k=−n−1 ⁿ ξ _k ⁽²ⁿ⁺¹⁾ z ^k , ξ _−n−1⁽²ⁿ⁺¹⁾>0. Associated with the even degree and odd degree OLPs are the following two pairs of recurrence relations: z φ _2n (z)=c _2n ^♯ φ _2n−2(z)+b _2n ^♯ φ _2n−1(z)+a _2n ^♯ φ _2n (z)+b _2n+1 ^♯ φ _2n+1(z)+c _2n+2 ^♯ φ _2n+2(z) and z φ _2n+1(z)=b _2n+1 ^♯ φ _2n (z)+a _2n+1 ^♯ φ _2n+1(z)+b _2n+2 ^♯ φ _2n+2(z), where c ₀ ^♯ =b ₀ ^♯ =0, and c _2k ^♯ >0, k∈ℕ, and z ⁻¹ φ _2n+1(z)=γ _2n+1 ^♯ φ _2n−1(z)+β _2n+1 ^♯ φ _2n (z)+α _2n+1 ^♯ φ _2n+1(z)+β _2n+2 ^♯ φ _2n+2(z)+γ _2n+3 ^♯ φ _2n+3(z) and z ⁻¹ φ _2n (z)=β _2n ^♯ φ _2n−1(z)+α _2n ^♯ φ _2n (z)+β _2n+1 ^♯ φ _2n+1(z), where β ₀ ^♯ =γ ₁ ^♯ =0, β ₁ ^♯ >0, and γ _2l+1 ^♯ >0, l∈ℕ. Asymptotics in the double-scaling limit N,n→∞ such that N/n=1+o(1) of the coefficients of these two pairs of recurrence relations, Hankel determinant ratios associated with the real-valued, bi-infinite strong moment sequence , and the products of the (real) roots of the OLPs are obtained by formulating the even degree and odd degree OLP problems as matrix Riemann-Hilbert problems on ℝ, and then extracting the large-n behaviours by applying the non-linear steepest-descent method introduced in (Ann. Math. 137(2):295–368, [1993]) and further developed in (Commun. Pure Appl. Math. 48(3):277–337, [1995]) and (Int. Math. Res. Not. 6:285–299, [1997]).      

On multiresolution analysis of multiplicityd

In this paper we study some basic properties of multiresolution analysis of multiplicityd in several variables and discuss some examples related to the spaces of cardinal splines with respect to the unidiagonal or the crisscross partition of the plane. Furthermore, in analogy with [8], we show that if the scaling functions are compactly supported, then it is possible to find compactly supported mother wavelets _l ,l=1,...,2 ⁿ d–d, in such a way that the family {2 ^jn/2 _l (2 ^j x–v)} is a semiorthogonal basis ofL ² ( ⁿ ).     

The Balian-Low theorem and regularity of Gabor systems

For any positive real numbers A, B, and d satisfying the conditions , d>2, we construct a Gabor orthonormal basis for L²(ℝ), such that the generating function g∈L²(ℝ) satisfies the condition:∫_ℝ|g(x)|²(1+|x| ^A )/log ^d (2+|x|)dx < ∞ and .