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.     

A COMPACTLY SUPPORTED ORTHONORMAL WAVELET OF NON-TENSOR TYPE

Abstract. In this note we construct an example of compactly supported orthonormal wavelets of non-tensor type from a multiresolutlon of     

Almost everywhere convergence of the inverse spherical transform onSL(2,R)

We prove the almost everywhere convergence of the inverse spherical transform ofL _p bi-K-invariant functions on the groupSL(2,R), 4/3<p≤2. The result appears to be sharp. Partially supported by the MPI Partially supported by the CMA     

Unitary mappings between multiresolution analyses of L2 (R) and a parametrization of low-pass filters

This article provides classes of unitary operators of L²(R) contained in the commutant of the Shift operator, such that for any pair of multiresolution analyses of L²(R) there exists a unitary operator in one of these classes, which maps all the scaling functions of the first multiresolution analysis to scaling functions of the other. We use these unitary operators to provide an interesting class of scaling functions. We show that the Dai-Larson unitary parametrization of orthonormal wavelets is not suitable for the study of scaling functions. These operators give an interesting relation between low-pass filters corresponding to scaling functions, which is implemented by a special class of unitary operators acting on L²([−π, π)), which we characterize. Using this characterization we recapture Daubechies' orthonormal wavelets bypassing the spectral factorization process. Acknowledgements and Notes. Partially supported by NSF Grant DMS-9157512, and Linear Analysis and Probability Workshop, Texas A&M University Dedicated to the memory of Professor Emeritus Vassilis Metaxas.     

Wavelet sets in ℝ n

A congruency theorem is proven for an ordered pair of groups of homeomorphisms of a metric space satisfying an abstract dilation-translation relationship. A corollary is the existence of wavelet sets, and hence of single-function wavelets, for arbitrary expansive matrix dilations on L ² (ℝ ⁿ). Moreover, for any expansive matrix dilation, it is proven that there are sufficiently many wavelet sets to generate the Borel structure ofℝ ⁿ. The second author is supported in part by NSF Grant DMS-9401544. The third author was a Graduate Research Assistant at Workshop in Linear Analysis and Probability, Texas A&M University.     

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.     

Characterization of Sobolev spaces of arbitrary smoothness using nonstationary tight wavelet frames

In this paper we shall characterize Sobolev spaces of an arbitrary order of smoothness using nonstationary tight wavelet frames for L ₂(ℝ). In particular, we show that a Sobolev space of an arbitrary fixed order of smoothness can be characterized in terms of the weighted ℓ₂-norm of the analysis wavelet coefficient sequences using a fixed compactly supported nonstationary tight wavelet frame in L ₂(ℝ) derived from masks of pseudosplines in [15]. This implies that any compactly supported nonstationary tight wavelet frame of L ₂(ℝ) in [15] can be properly normalized into a pair of dual frames in the corresponding pair of dual Sobolev spaces of an arbitrary fixed order of smoothness. Research supported in part by NSERC Canada under Grant RGP 228051. Research supported in part by Grant R-146-000-060-112 at the National University of Singapore.     

A lower bound for the number of singular staturated main effect plans of anS m factorial

Summary A fraction consisting ofm(s−1)+1 observations taken atm(s−1)+1 treatment combinations of ans ^m factorial with the aim to estimate the mean and them(s−1), main effect single degree of freedom parameters is called a saturated main effect plan. If the design matrix of such a fraction is singular, then the fraction is called a singular saturated main effect plan. This paper presents a lower bound on the number of singular saturated main effect plans. Research supported by N.R.C. Grant No. A07204.     

Almost everywhere convergence of Riesz means on noncompact symmetric spaceSL(3,H)/Sp(3)

LetG/K be the noncompact Riemannian symmetric spaceSL(3,H)/Sp(3). We shall prove in this paper that forf∈L ^p(SL(3,H)/Sp(3)), 1≤p≤2, the Riesz means of orderz off with respect to the eigenfunctions expansion of Laplace operator almost everywhere converge tof for Rez >δ(n,p). The critical index δ(n,p) is the same as in the classical Stein's result for Euclidean space, and as in the noncompact symmetric spaces of rank one and of complex type. Partially supported by National Natural Science Foundation of China     

A study on orthogonal two-direction vector-valued wavelets and two-direction wavelet packets

In this paper, the notion of two-direction vector-valued multiresolution analysis and the two-direction orthogonal vector-valued wavelets are introduced. The definition for two-direction orthogonal vector-valued wavelet packets is proposed. An algorithm for constructing a class of two-direction orthogonal vector-valued compactly supported wavelets corresponding to the two-direction orthogonal vector-valued compactly supported scaling functions is proposed by virtue of matrix theory and time-frequency analysis method. The properties of the two-direction vector-valued wavelet packets are investigated. At last, the direct decomposition relation for space L₂(R)^r is presented.     

Radicals of crossed products of enveloping algebras

LetL be a Lie algebra over a fieldK which acts asK-derivations on aK-algebraR. Then this action determines a crossed productR *U(L) whereU(L) is the enveloping algebra ofL. The goal of this paper is to describe the Jacobson radical ofR * U(L) forL≠0. We are most successful whenR is a p.i. algebra or Noetherian. In more general situations we at least obtain upper and lower bounds forJ(R * U(L)) which are ideals extended fromR. Furthermore, we offer an interesting example in all characteristics of a commutativeK-algebraC which admits a derivationδ such thatC isδ-prime but not semiprime. Partially supported by N.S.F. Grant No. DMS 85-00959 and by a Guggenheim Memorial Foundation Fellowship. Partially supported by N.S.F. Grant No. MCS 82-19678.     

Cayley’s decomposition and pólya’sW-property of ordinary linear differential equations

A proof is given for the equivalence of Pólya’sW- property of a linear differential equationL _n (D) y=0 to the possibility of decomposingL _n (D) ≡ Π _n ¹ [D+λ_i(x)] in a given interval. In this case a set ofn independent solutions form a Chebyshev system in the interval. An application determines intervals of non-oscillation for solutions of linear equations of the second order. Research supported by the National Science Foundation Grant No. GP-3897 with the University of Maryland.     

The Calderón reproducing formula, windowed X-ray transforms, and radon transforms in LP-spaces

The generalized Calderón reproducing formula involving “wavelet measure” is established for functions f ∈ L^p(ℝⁿ). The special choice of the wavelet measure in the reproducing formula gives rise to the continuous decomposition of f into wavelets, and enables one to obtain inversion formulae for generalized windowed X-ray transforms, the Radon transform, and k-plane transforms. The admissibility conditions for the wavelet measure μ are presented in terms of μ itself and in terms of the Fourier transform of μ. Acknowledgements and Notes. Partially sponsored by the Edmund Landau Center for research in Mathematical Analysis, supported by the Minerva Foundation (Germany).     

Wavelet Approximation of Distributions with Bounded Variation Derivatives

Let BV ^r denote the space of distributions f such that the distributional derivatives D ^α f with |α|≤r exist as measures of bounded variation. This paper discusses estimates for wavelet coefficients of BV ^r distributions, direct (Jackson) and inverse (Bernstein) inequalities for n-term approximation of elements of BV ^r in the L _p spaces using compactly supported wavelets. In particular, optimal rates of approximation are established. Linear approximation in similar contexts is also considered for comparison. This research was supported by the 2003–2007 Academic Grant of Prof. P. Wojtaszczyk awarded by the Foundation for Polish Science. Part of this research was supported within the HASSIP framework.     

Deformation retracts and the Hochschild homology of polynomial rings

We assume given a ringA with unit, and a subcomplex of the reduced bar complex ofA. We assume that this subcomplex is a deformation retract of the whole complex and thus has homology equal to the Hochschild homology ofA, but it will typically be smaller and easier to calculate with. We use these to construct (accordingly small) deformation retracts for the reduced bar complexes ofA[t] andA[t,t ⁻¹]. WhenA is a Banach algebra, we also do this construction forC ^∞(S¹;A). Partially supported by N.S.F. Grant No. DMS 92-03398.     

A general construction of nonseparable multivariate orthonormal wavelet bases

The construction of nonseparable and compactly supported orthonormal wavelet bases of L ²(R ⁿ ); n ≥ 2, is still a challenging and an open research problem. In this paper, we provide a special method for the construction of such wavelet bases. The wavelets constructed by this method are dyadic wavelets. Also, we show that our proposed method can be adapted for an eventual construction of multidimensional orthogonal multiwavelet matrix masks, candidates for generating multidimensional multiwavelet bases.      

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.     

On the galois module structure of semisimple holomorphic differentials

LetL/K be a finite Galoisp-extension of algebraic function fields of one variable over an algebraically closed fieldk of characteristicp, with Galois groupG=Gal(L/K). The space Ώ _L ^s (0) of semisimple holomorphic differentials ofL is thek-vector space of holomorphic differentials which are fixed by the Cartier operator. We obtain the isomorphism classes and multiplicities of the summands in a Krull-Schmidt decomposition of thek[G]-module Ώ _L ^s (0) into a direct sum of indecomposablek[G]-modules. Partially supported by CONACyT, project No. 25063-E.     

A study on compactly supported orthogonal vector-valued wavelets and wavelet packets

In this paper, vector-valued multiresolution analysis and orthogonal vector-valued wavelets are introduced. The definition for orthogonal vector-valued wavelet packets is proposed. A necessary and sufficient condition on the existence of orthogonal vector-valued wavelets is derived by means of paraunitary vector filter bank theory. An algorithm for constructing a class of compactly supported orthogonal vector-valued wavelets is presented. The properties of the vector-valued wavelet packets are investigated by using operator theory and algebra theory. In particular, it is shown how to construct various orthonormal bases of L²(R, C^s) from the orthogonal vector-valued wavelet packets.     

Quaternion-valued admissible wavelets and orthogonal decomposition of L 2(IG(2), ℍ)

A series of admissible wavelets is fixed, which forms an orthonormal basis for the Hilbert space of all the quaternion-valued admissible wavelets. It turns out that their corresponding admissible wavelet transforms give an orthogonal decomposition of L ²(IG(2), ℍ).