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.     

Approximation order provided by refinable function vectors

In this paper we considerL _p-approximation by integer translates of a finite set of functionsϕ _v (v=0, ...,r − 1) which are not necessarily compactly supported, but have a suitable decay rate. Assuming that the function vectorϕ=(ϕ ^=0/ r−1 is refinable, necessary and sufficient conditions for the refinement mask are derived. In particular, if algebraic polynomials can be exactly reproduced by integer translates ofϕ _v, then a factorization of the refinement mask ofϕ can be given. This result is a natural generalization of the result for a single functionϕ, where the refinement mask ofϕ contains the factor ((1 +e ^−iu )/2) ^m if approximation orderm is achieved. Dedicated to Professor L. Berg on the occasion of his 65th birthday     

Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix

In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space W_p^k(ℝ^s) (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 W_p^k(ℝ^s) will be investigated. When the dilation matrix is isotropic, a characterization will be given for the L_p (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∈L_p(ℝ^s) (φ∈C(ℝ^s) 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,L_p(ℝ^s)) 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.     

Refinement equations with nonnegative coefficients

In this paper we analyze solutions of the n-scale functional equation Ф(x) = Σ_k∈ℤ ^Pk Ф(nx−k), where n≥2 is an integer, the coefficients {P_k} are nonnegative and Σpk = 1. We construct a sharp criterion for the existence of absolutely continuous solutions of bounded variation. This criterion implies several results concerning the problem of integrable solutions of n-scale refinement equations and the problem of absolutely continuity of distribution function of one random series. Further we obtain a complete classification of refinement equations with positive coefficients (in the case of finitely many terms) with respect to the existence of continuous or integrable compactly supported solutions.     

A deterministic scheme for Smoluchowski's coagulation equation based on binary grid refinement

We present a deterministic scheme for the discrete Smoluchowski's coagulation equation based on a binary grid refinement. Starting from the binary grid Ω₀={1,2,4,8,16,. . .}, we first introduce an appropriate grid refinement by adding at each level 2^l grid points in every binary subsection of the grid Ω_l. In a next step we derive an approximate equation for the dynamic behavior on each level Ω_l based on a piecewise constant approximation of the right hand side of Smoluchowski's equation. Numerical results show that the computational effort can be drastically decreased compared to the corresponding complete integer grid. When considering unbounded kernels in Smoluchowski's equation we use an adaptive time step method to overcome numerical instabilities which may occur at the tails of the density function.     

Stability estimates for h-p spectral element methods for elliptic problems

In a series of papers of which this is the first we study how to solve elliptic problems on polygonal domains using spectral methods on parallel computers. To overcome the singularities that arise in a neighborhood of the corners we use a geometrical mesh. With this mesh we seek a solution which minimizes a weighted squared norm of the residuals in the partial differential equation and a fractional Sobolev norm of the residuals in the boundary conditions and enforce continuity by adding a term which measures the jump in the function and its derivatives at inter-element boundaries, in an appropriate fractional Sobolev norm, to the functional being minimized. Since the second derivatives of the actual solution are not square integrable in a neighborhood of the corners we have to multiply the residuals in the partial differential equation by an appropriate power of r_k, where r_k measures the distance between the pointP and the vertexA _k in a sectoral neighborhood of each of these vertices. In each of these sectoral neighborhoods we use a local coordinate system (τ_k, θ_k) where τ_k = lnr_k and (r_k, θ_k) are polar coordinates with origin at A_k, as first proposed by Kondratiev. We then derive differentiability estimates with respect to these new variables and a stability estimate for the functional we minimize. In [6] we will show that we can use the stability estimate to obtain parallel preconditioners and error estimates for the solution of the minimization problem which are nearly optimal as the condition number of the preconditioned system is polylogarithmic inN, the number of processors and the number of degrees of freedom in each variable on each element. Moreover if the data is analytic then the error is exponentially small inN.     

Kalai-Smorodinsky Bargaining Solution Equilibria

Multicriteria games describe strategic interactions in which players, having more than one criterion to take into account, don’t have an a-priori opinion on the relative importance of all these criteria. Roemer (Econ. Bull. 3:1–13, 2005) introduces an organizational interpretation of the concept of equilibrium: each player can be viewed as running a bargaining game among criteria. In this paper, we analyze the bargaining problem within each player by considering the Kalai-Smorodinsky bargaining solution (see Kalai and Smorodinsky in Econometrica 43:513–518, 1975). We provide existence results for the so called Kalai-Smorodinsky bargaining solution equilibria for a general class of disagreement points which properly includes the one considered by Roemer (Econ. Bull. 3:1–13, 2005). Moreover we look at the refinement power of this equilibrium concept and show that it is an effective selection device even when combined with classical refinement concepts based on stability with respect to perturbations; in particular, we consider the extension to multicriteria games of the Selten’s trembling hand perfect equilibrium concept (see Selten in Int. J. Game Theory 4:25–55, 1975) and prove that perfect Kalai-Smorodinsky bargaining solution equilibria exist and properly refine both the perfect equilibria and the Kalai-Smorodinsky bargaining solution equilibria.     

The spectral radius of matrix continuous refinement operators

A simple analytic formula for the spectral radius of matrix continuous refinement operators is established. On the space L₂^m(\mathbb R^s)L_2^m({{\mathbb R}}^s), m ≥ 1 and s ≥ 1, their spectral radius is equal to the maximal eigenvalue in magnitude of a number matrix, obtained from the dilation matrix M and the matrix function c defining the corresponding refinement operator. A similar representation is valid for the continuous refinement operators considered on spaces L _p for p ∈ [1, ∞ ), p ≠ 2. However, additional restrictions on the kernel c are imposed in this case.     

Refinable Functions with Exponential Decay: An?Approach via Cascade Algorithms

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.     

A finite volume scheme for nonlinear parabolic equations derived from one-dimensional local dirichlet problems

In this paper, we propose a new method to compute the numerical flux of a finite volume scheme, used for the approximation of the solution of the nonlinear partial differential equation u_t+div(qf(u))−ΔΦ(u)=0 in a 1D, 2D or 3D domain. The function Φ is supposed to be strictly increasing, but some values s such that Φ′(s)=0 can exist. The method is based on the solution, at each interface between two control volumes, of the nonlinear elliptic two point boundary value problem (qf(υ)+(Φ(υ))′)′=0 with Dirichlet boundary conditions given by the values of the discrete approximation in both control volumes. We prove the existence of a solution to this two point boundary value problem. We show that the expression for the numerical flux can be yielded without referring to this solution. Furthermore, we prove that the so designed finite volume scheme has the expected stability properties and that its solution converges to the weak solution of the continuous problem. Numerical results show the increase of accuracy due to the use of this scheme, compared to some other schemes.     

On subadditive functions and ψ-additive mappings

In [4], assuming among others subadditivity and submultiplicavity of a function ψ: [0, ∞)→[0, ∞), the authors proved a Hyers-Ulam type stability theorem for “ψ-additive” mappings of a normed space into a normed space. In this note we show that the assumed conditions of the function ψ imply that ψ=0 and, consequently, every “ψ-additive” mapping must be additive     

Fourier method in an initial-boundary value problem for a first-order partial differential equation with involution

The Fourier method is used to obtain a classical solution of an initial-boundary value problem for a first-order partial differential equation with involution in the function and its derivative. The series Σ produced by the Fourier method as a formal solution of the problem is represented as Σ = S ₀ + (Σ − Σ₀₎, where Σ₀ is the formal solution of a special reference problem for which the sum S ₀ can be explicitly calculated. Refined asymptotic formulas for the solution of the Dirac system are used to show that the series Σ − Σ₀ and the series obtained from it by termwise differentiation converge uniformly. Minimal smoothness assumptions are imposed on the initial data of the problem.     

p-Adic Haar Multiresolution Analysis and Pseudo-Differential Operators

The notion of p-adic multiresolution analysis (MRA) is introduced. We discuss a “natural” refinement equation whose solution (a refinable function) is the characteristic function of the unit disc. This equation reflects the fact that the characteristic function of the unit disc is a sum of p characteristic functions of mutually disjoint discs of radius p ⁻¹. This refinement equation generates a MRA. The case p=2 is studied in detail. Our MRA is a 2-adic analog of the real Haar MRA. But in contrast to the real setting, the refinable function generating our Haar MRA is 1-periodic, which never holds for real refinable functions. This fact implies that there exist infinity many different 2-adic orthonormal wavelet bases in ℒ²(ℚ₂) generated by the same Haar MRA. All of these new bases are described. We also constructed infinity many different multidimensional 2-adic Haar orthonormal wavelet bases for ℒ²(ℚ₂ ⁿ ) by means of the tensor product of one-dimensional MRAs. We also study connections between wavelet analysis and spectral analysis of pseudo-differential operators. A criterion for multidimensional p-adic wavelets to be eigenfunctions for a pseudo-differential operator (in the Lizorkin space) is derived. We proved also that these wavelets are eigenfunctions of the Taibleson multidimensional fractional operator. These facts create the necessary prerequisites for intensive using our wavelet bases in applications. Our results related to the pseudo-differential operators develop the investigations started in Albeverio et al. (J. Fourier Anal. Appl. 12(4):393–425, 2006).      

Convergence of an explicit finite volume scheme for first order symmetric systems

Summary.  This paper is devoted to the derivation of a O(h ^1/2) error estimate for the classical upwind, explicit in time, finite volume scheme for linear first order symmetric systems. Such a result already existed for the corresponding implicit in time finite volume scheme, since it can be interpreted as a particular case of the space-time discontinuous Galerkin method but the technique of proof, used in that case, does not extend to explicit schemes. The general framework, recently developed to analyse the convergence rate of finite volume schemes for non linear scalar conservation laws, can not be used either, because it is not adapted for systems, even linear. In this article, we propose a new technique, which takes advantage of the linearity of the problem. The first step consists in controlling the approximation error ∥u−u _h ∥ _L2 by an expression of the form <ν _h , g>−2<μ _h , gu>, where u is the exact solution, g is a particular smooth function, and μ _h , ν _h are some linear forms depending on the approximate solution u _h . The second step consists in carefully estimating the error terms <μ _h , gu> and <ν _h , g>, by using uniform stability results for the discrete problem and regularity properties of the continuous solution. Received December 20, 2001 / Revised version received January 2, 2001 / Published online November 27, 2002 Mathematics Subject Classification (1991): 65N30     

Regularity of refinable function vectors

We study the existence and regularity of compactly supported solutions φ = (φ_v) _v=0 ^/r−1 of vector refinement equations. The space spanned by the translates of φ_v can only provide approximation order if the refinement maskP has certain particular factorization properties. We show, how the factorization ofP can lead to decay of |̸_v(u)| as |u| → ∞. The results on decay are used to prove uniqueness of solutions and convergence of the cascade algorithm.     

Compactly Supported Distributional Solutions of Nonstationary Nonhomogeneous Refinement Equations

Let A be a matrix with the absolute values of all eigenvalues strictly larger than one, and let Z ₀ be a subset of Z such than n∈Z ₀ implies n + 1 ∈Z ₀. 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 G _n and H _n , n∈Z ₀, in D′, define the corresponding nonstationary nonhomogeneous refinement equation Φ _n =H _n *Φ _n+1 (A·)+G _n for all n∈Z ₀ where Φ _n , n∈Z ₀, 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 Φ _n , n∈Z ₀, of the equation (*). In fact, we reduce the existence problem to finding a bounded solution of the linear equations for all n∈Z ₀ where the matrices S _n and the vectors , n∈Z ₀, can be constructed explicitly from H _n and G _n respectively. The results above are still new even for stationary nonhomogeneous refinement equations. Received December 30, 1999, Accepted June 15, 2000     

On Existence and Weak Stability of Matrix Refinable Functions

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.     

Pointwise fourier inversion: A wave equation approach

Functions of the Laplace operator F(− Δ) can be synthesized from the solution operator to the wave equation. When F is the characteristic function of [0, R ² ], this gives a representation for radial Fourier inversion. A number of topics related to pointwise convergence or divergence of such inversion, as R → ∞, are studied in this article. In some cases, including analysis on Euclidean space, sphers, hyperbolic space, and certain other symmetric spaces, exact formulas for fundamental solutions to wave equations are available. In other cases, parametrices and other tools of microlocal analysis are effective.     

Localized Polynomial Frames on the Interval with Jacobi Weights

As is well known the kernel of the orthogonal projector onto the polynomials of degree n in L²(w_α,β, [−1, 1]), w_α,β(t) = (1 − t)^α(1 + t)^β, can be written in terms of Jacobi polynomials. It is shown that if the coefficients in this kernel are smoothed out by sampling a compactly supported C^∞ function then the resulting function has nearly exponential (faster than any polynomial) rate of decay away from the main diagonal. This result is used for the construction of tight polynomial frames for L²(w_α,β) with elements having almost exponential localization.     

A boundary-value problem for the biharmonic equation and the iterated Laplacian in a 3D-domain with an edge

Let Ω be a domain with piecewise smooth boundary. In general, it is impossible to obtain a generalized solution u ∈ W ₂ ² (Ω) of the equation Δ _x ² u = f with the boundary conditions u = Δ_xu = 0 by solving iteratively a system of two Poisson equations under homogeneous Dirichlet conditions. Such a system is obtained by setting v = −Δu. In the two-dimensional case, this fact is known as the Sapongyan paradox in the theory of simply supported polygonal plates. In the present paper, the three-dimensional problem is investigated for a domain with a smooth edge Γ. If the variable opening angle α ∈ C^∞(Γ) is less than π everywhere on the edge, then the boundary-value problem for the biharmonic equation is equivalent to the iterated Dirichlet problem, and its solution u inherits the positivity preserving property from these problems. In the case α ∈ (π 2π), the procedure of solving the two Dirichlet problems must be modified by permitting infinite-dimensional kernel and co-kernel of the operators and determining the solution u ∈ W ₂ ² (Ω) by inverting a certain integral operator on the contour Γ. If α(s) ∈ (3π/2,2π) for a point s ∈ Γ, then there exists a nonnegative function f ∈ L₂(Ω) for which the solution u changes sign inside the domain Ω. In the case of crack (α = 2π everywhere on Γ), one needs to introduce a special scale of weighted function spaces. In this case, the positivity preserving property fails. In some geometrical situations, the problems on well-posedness for the boundary-value problem for the biharmonic equation and the positivity property remain open. Bibliography: 46 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 336, 2006, pp. 153–198.