On some class of nonseparable continuous wavelet transforms

It is well-known that only a single condition (called the admissibility condition) is sufficient for L ₂-convergence of multiple continuous wavelet transforms (MCWT). However known results suggest that to guarantee the pointwise convergence of MCWT for L _p -functions wavelets should vanish quite rapidly at infinity. In this article, we consider the class of nonseparable multiple wavelets with square-symmetric Fourier transforms. For these wavelets ψ we prove that the admissibility condition and the condition ψ∈L ₁(R ⁿ ) are sufficient for the convergence of corresponding MCWT almost everywhere.     

Boundedness of Convolution-Type Operators on Certain Endpoint Triebel-Lizorkin Spaces

In this paper, we are concerned with the boundedness of convolution-type Calderón-Zygmund operators on some endpoint Triebel-Lizorkin spaces. We establish the boundedness on [(F)\dot]₁^0,q\dot{F}_{1}^{0,q} (2<q<∞) under a very weak pointwise regularity condition. The boundedness is established by the Daubechies wavelets and the atomic-molecular approach.     

Homogeneous Approximation Property for Multivariate Continuous Wavelet Transforms

The homogeneous approximation property (HAP) for the continuous wavelet transform is useful in practice because it means that the measure of the building area involved in a reconstruction of a function up to some error is essentially invariant under timescale shifts. For the univariate case, it was shown that the pointwise HAP holds if and only if the Fourier transforms of both wavelets and the function to be reconstructed are compactly supported on ??{0}. In this paper, we study the HAP for multivariate wavelet transforms. We show that similar results hold for this case. However, the above condition is only sufficient but not necessary if the dimension of the variable is greater than 1, which is different from the univariate case. We also get a convergence result on the inverse of wavelet transforms, which improves similar results by Daubechies and Holschneider and Tchamitchain.     

Pointwise Green's function approach to stability for scalar conservation laws

We study the pointwise behavior of perturbations from a viscous shock solution to a scalar conservation law, obtaining an estimate independent of shock strength. We find that for a perturbation with initial data decaying algebraically or slower, the perturbation decays in time at the rate of decay of the integrated initial data in any L^p norm, p ≥ 1. Stability in any L^p norm is a direct consequence. The approach taken is that of obtaining pointwise estimates on the perturbation through a Duhamel's principle argument that employs recently developed pointwise estimates on the Green's function for the linearized equation. © 1999 John Wiley & Sons, Inc.     

Smoothing minimally supported frequency wavelets: Part II

The main purpose of this paper is to give a procedure to “mollify” the low-pass filters of a large number ofMinimally Supported Frequency (MSF) wavelets so that the smoother functions obtained in this way are also low-pass filters for an MRA. Hence, we are able to approximate (in the L ² -norm) MSF wavelets by wavelets with any desired degree of smoothness on the Fourier transform side. Although the MSF wavelets we consider are bandlimited, this may not be true for their smooth approximations. This phenomena is related to the invariant cycles under the transformation x ↦2x (mod2π). We also give a characterization of all low-pass filters for MSF wavelets. Throughout the paper new and interesting examples of wavelets are described.     

Benchmarking of Three-Dimensional Clifford Wavelet Functions

Specific three dimensional wavelet functions for the Continuous Wavelet Transform (CWT) in the framework of Clifford analysis are tested upon their selectivity for pointwise and directional analysis of signals. These tests are carried out on scalar benchmark signals: the infinite rod, the semi-infinite rod and the rod of finite length. It is demonstrated that the Clifford-Hermite wavelets are efficient at detecting pointwise singularities, whereas the Clifford-Morlet wavelets possess the qualifications for directional analysis of three-dimensional signals.     

Wavelet Approximation of Periodic Functions

We investigate expansions of periodic functions with respect to wavelet bases. Direct and inverse theorems for wavelet approximation in C and L_p norms are proved. For the functions possessing local regularity we study the rate of pointwise convergence of wavelet Fourier series. We also define and investigate the “discreet wavelet Fourier transform” (DWFT) for periodic wavelets generated by a compactly supported scaling function. The DWFT has one important advantage for numerical problems compared with the corresponding wavelet Fourier coefficients: while fast computational algorithms for wavelet Fourier coefficients are recursive, DWFTs can be computed by explicit formulas without any recursion and the computation is fast enough.     

Maximal L 2 and Pointwise Hölder Estimates for □ b on CR Manifolds of Class C 2

In this paper we prove the optimal pointwise Hölder estimates for the Kohn's Laplacian □ _b for CR manifolds of class C ². We first derive optimal L ² theory for solutions of □ _b directly without using pseudo-differential operators. Then we use the scaling method to obtain pointwise estimates in non-istotropic Campanato spaces based on the maximal L ² theory.     

Pointwise convergence of ergodic averages along cubes

Let (X, B, μ, T) be a measure preserving system. We prove the pointwise convergence of ergodic averages along cubes of 2 ^k − 1 bounded and measurable functions for all k. We show that this result can be derived from estimates about bounded sequences of real numbers and apply these estimates to establish the pointwise convergence of some weighted ergodic averages and ergodic averages along cubes for not necessarily commuting measure preserving transformations.     

Large classes of minimally supported frequency wavelets of<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(ℝ) and<Emphasis Type="Italic">H</Emphasis><Superscript>2</Superscript>(ℝ)

We introduce a new method to construct large classes of minimally supported frequency (MSF) wavelets of the Hardy space H ² (ℝ)and symmetric MSF wavelets of L ² (ℝ),and discuss the classification of such wavelets. As an application, we show that there are uncountably many such wavelet sets of L ² (ℝ)and H ² (ℝ).We also enumerate some of the symmetric wavelet sets of L ² (ℝ)and all wavelet sets of H ² (ℝ)consisting of three intervals. Finally, we construct families of MSF wavelets of L ₂ (ℝ)with Fourier transform even and not vanishing in any neighborhood of the origin.     

Construction of optimally conditioned cubic spline wavelets on the interval

The paper is concerned with a construction of new spline-wavelet bases on the interval. The resulting bases generate multiresolution analyses on the unit interval with the desired number of vanishing wavelet moments for primal and dual wavelets. Both primal and dual wavelets have compact support. Inner wavelets are translated and dilated versions of well-known wavelets designed by Cohen, Daubechies, and Feauveau. Our objective is to construct interval spline-wavelet bases with the condition number which is close to the condition number of the spline wavelet bases on the real line, especially in the case of the cubic spline wavelets. We show that the constructed set of functions is indeed a Riesz basis for the space L ² ([0, 1]) and for the Sobolev space H ^s ([0, 1]) for a certain range of s. Then we adapt the primal bases to the homogeneous Dirichlet boundary conditions of the first order and the dual bases to the complementary boundary conditions. Quantitative properties of the constructed bases are presented. Finally, we compare the efficiency of an adaptive wavelet scheme for several spline-wavelet bases and we show a superiority of our construction. Numerical examples are presented for the one-dimensional and two-dimensional Poisson equations where the solution has steep gradients.     

Complete Riemannian manifolds with pointwise pinched curvature

An analogous Bonnet-Myers theorem is obtained for a complete and positively curved n-dimensional (n≥3) Riemannian manifold M ⁿ . We prove that if n≥4 and the curvature operator of M ⁿ is pointwise pinched, or if n=3 and the Ricci curvature of M ³ is pointwise pinched, then M ⁿ is compact. Oblatum 4-II-1999 & 10-XI-1999?Published online: 21 February 2000     

Pointwise Error Estimate for Spectral Galerkin Approximations of Micropolar Equations

The goal of this article is to present pointwise time error estimates in suitable Hilbert spaces by considering spectral Galerkin approximations of the micropolar fluid model for strong solutions. In fact, we use the properties of the Stokes and Lamé operators for prove the pointwise convergence rate in the H²-norm for the ordinary velocity and microrotational velocity and the pointwise convergence rate in the L²-norm for the time-derivative of both velocities. The novelty of our method is that we do not impose any compatibility conditions in the initial data.     

ON CONVERGENCE OF WAVELET PACKET EXPANSIONS

It is well known that the-Walsh-Fourier expansion of a function from the block spaceB _q([0,1]), 1B _q in certain periodized smooth periodic non-stationary wavelet packets bases based on the Haar filters. We also consider wavelet packets based on the Shannon filters and show that the expansion of L^p-functions, 1相似文献   

Random interval graphs

We consider models for random interval graphs that are based on stochastic service systems, with vertices corresponding to customers and edges corresponding to pairs of customers that are in the system simultaneously. The number N of vertices in a connected component thus corresponds to the number of customers arriving during a busy period, while the size K of the largest clique (which for interval graphs is equal to the chromatic number) corresponds to the maximum number of customers in the system during a busy period. We obtain the following results for both the M/D/∞ and the M/M/∞ models, with arrival rate λ per mean service time. The expected number of vertices is e^λ, and the distribution of the N/e^λ converges pointwise to an exponential distribution with mean 1 as λ tends to infinity. This implies that the distribution of log N−λ converges pointwise to a distribution with mean −γ (where γ is Euler's constant) and variance π²/6. The size K of the largest clique falls in the interval [eλ−2 log λ, eλ+1] with probability tending to 1 as λ tends to infinity. Thus the distribution of the ratio K/log N converges pointwise to that of the constant e, in contrast to the situation for random graphs generated by unbiased coin flips, in which the distribution of K/log N converges pointwise to that of the constant 2/log 2. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12: 361–380, 1998     

A New Method of Constructions of Non-Tensor Product Wavelets

In this paper, we give a new method of constructions of non-tensor product wavelets. We start from the one-dimensional scaling functions to directly construct the two-dimensional non-tensor product wavelets. The wavelets constructed by us possess very simple, explicit representations and high regularity, and various symmetry (i.e., axial symmetry, central symmetry, and cyclic symmetry). Using this method, we construct various non-tensor product wavelets and show that there exists a sequence of non-tensor product wavelets with high regularity which tends to the tensor product Shannon wavelet in the L ²-norm.     

Discrete Radon transforms and applications to ergodic theory

We prove L ^p boundedness of certain non-translation-invariant discrete maximal Radon transforms and discrete singular Radon transforms. We also prove maximal, pointwise, and L ^p ergodic theorems for certain families of non-commuting operators.     

Locally Supported,Piecewise Polynomial Biorthogonal Wavelets on Nonuniform Meshes

In this paper, biorthogonal wavelets are constructed on nonuniform meshes. Both primal and dual wavelets are locally supported, continuous piecewise polynomials. The wavelets generate Riesz bases for the Sobolev spaces (H^s) for (|s| < 3/2). The wavelets at the primal side span standard Lagrange finite element spaces.     

On the (x,t) asymptotic properties of solutions of the Navier-Stokes equations in the half-space

We study the space-time asymptotic behavior of classical solutions of the initial-boundary value problem for the Navier-Stokes system in the half-space. We construct a (local in time) solution corresponding to an initial data that is only assumed to be continuous and decreasing at infinity as |x|^−μ, μ ∈ (1/2,n). We prove pointwise estimates in the space variable. Moreover, if μ ∈ [1, n) and the initial data is suitably small, then the above solutions are global (in time), and we prove space-time pointwise estimates. Bibliography: 19 titles. Alla memoria di Olga Aleksandrovna Ladyzhenskaya Published in Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 147–202.     

On the Riemann Summability of Fourier Integrals and Real Hardy Spaces

We consider the Riemann means of single and multiple Fourier integrals of functions belonging to L¹ or the real Hardy spaces defined on ℝⁿ, where n ≥ 1 is an integer. We prove that the maximal Riemann operator is bounded both from H¹(ℝ) into L¹(ℝ) and from L¹(ℝ) into weak –L¹(ℝ). We also prove that the double maximal Riemann operator is bounded from the hybrid Hardy spaces H^(1,0)(ℝIsup2), H^(0,1)(ℝ²⁾ into weak –L¹(ℝ²). Hence pointwise Riemann summability of Fourier integrals of functions in H^(1,0) ∪ H^(0,1)(ℝ²) follows almost everywhere.The maximal conjugate Riemann operators as well as the pointwise convergence of the conjugate Riemann means are also dealt with.