Band-Limited Wavelets

Using the harmonic method,we get a class of more general band-limited wavelet,which was got by complicate operator interpolation method before.Our result is alittle better than the result by operator interpolation method.The fast band-limitedwavelet transform shall be given in another paper.     

Spectral Models for Orthonormal Wavelets and?Multiresolution Analysis of L 2(?)

Spectral representations of the dilation and translation operators on L ²(?) are built through appropriate bases. Orthonormal wavelets and multiresolution analysis are then described in terms of rigid operator-valued functions defined on the functional spectral spaces. The approach is useful for computational purposes.     

Pointwise characterization for combinations of Szász-Mirakjan operators

The purpose of this paper is to characterize the pointwise rate of convergence for the combinations of Szász-Mirakjan operators using Ditzian-Totik modulus of smoothness.     

Pointwise characterization for combinations of Szász-Mirakjan operators

The purpose of this paper is to characterize the pointwise rate of convergence for the combinations of Szász-Mirakjan operators using Ditzian-Totik modulus of smoothness.     

A Smirnov–Bickel–Rosenblatt Theorem for Compactly-Supported Wavelets

In nonparametric statistical problems, we wish to find an estimator of an unknown function f. We can split its error into bias and variance terms; Smirnov, Bickel and Rosenblatt have shown that, for a histogram or kernel estimate, the supremum norm of the variance term is asymptotically distributed as a Gumbel random variable. In the following, we prove a version of this result for estimators using compactly-supported wavelets, a popular tool in nonparametric statistics. Our result relies on an assumption on the nature of the wavelet, which must be verified by provably-good numerical approximations. We verify our assumption for Daubechies wavelets and symlets, with N=6,…,20 vanishing moments; larger values of N, and other wavelet bases, are easily checked, and we conjecture that our assumption holds also in those cases.     

Pointwise multipliers for Besov spaces of dominating mixed smoothness-Ⅱ

We continue our investigations on pointwise multipliers for Besov spaces of dominating mixed smoothness. This time we study the algebra property of the classes S_(p,q)~rB(R~d) with respect to pointwise multiplication. In addition, if p≤q, we are able to describe the space of all pointwise multipliers for S_(p,q)~rB(R~d).     

Averaged Pointwise Bounds for Deformations of Schrödinger Eigenfunctions

Let (M,g) be an n-dimensional, compact Riemannian manifold and ${P_0(\hbar) = -\hbar{^2} \Delta_g + V(x)}$ be a semiclassical Schrödinger operator with ${\hbar \in (0,\hbar_0]}$ . Let ${E(\hbar) \in [E-o(1),E+o(1)]}$ and ${(\phi_{\hbar})_{\hbar \in (0,\hbar_0]}}$ be a family of L ²-normalized eigenfunctions of ${P_0(\hbar)}$ with ${P_0(\hbar) \phi_{\hbar} = E(\hbar) \phi_{\hbar}}$ . We consider magnetic deformations of ${P_0(\hbar)}$ of the form ${P_u(\hbar) = - \Delta_{\omega_u}(\hbar) + V(x)}$ , where ${\Delta_{\omega_u}(\hbar) = (\hbar d + i \omega_u(x))^*({\hbar}d + i \omega_u(x))}$ . Here, u is a k-dimensional parameter running over ${B^k(\epsilon)}$ (the ball of radius ${\epsilon}$ ), and the family of the magnetic potentials ${(w_u)_{u\in B^k(\epsilon)}}$ satisfies the admissibility condition given in Definition 1.1. This condition implies that k ≥ n and is generic under this assumption. Consider the corresponding family of deformations of ${(\phi_{\hbar})_{\hbar \in (0, \hbar_0]}}$ , given by ${(\phi^u_{\hbar})_{\hbar \in(0, \hbar_0]}}$ , where $$\phi_{\hbar}^{(u)}:= {\rm e}^{-it_0 P_u(\hbar)/\hbar}\phi_{\hbar}$$ for ${|t_0|\in (0,\epsilon)}$ ; the latter functions are themselves eigenfunctions of the ${\hbar}$ -elliptic operators ${Q_u(\hbar): ={\rm e}^{-it_0P_u(\hbar)/\hbar} P_0(\hbar) {\rm e}^{it_0 P_u(\hbar)/\hbar}}$ with eigenvalue ${E(\hbar)}$ and ${Q_0(\hbar) = P_{0}(\hbar)}$ . Our main result, Theorem1.2, states that for ${\epsilon >0 }$ small, there are constants ${C_j=C_j(M,V,\omega,\epsilon) > 0}$ with j = 1,2 such that $$C_{1}\leq \int\limits_{\mathcal{B}^k(\epsilon)} |\phi_{\hbar}^{(u)}(x)|^2 \, {\rm d}u \leq C_{2}$$ , uniformly for ${x \in M}$ and ${\hbar \in (0,h_0]}$ . We also give an application to eigenfunction restriction bounds in Theorem 1.3.     

Frame Wavelets with Compact Supports for L^2（R^n）

The construction of frame wavelets with compact supports is a meaningful problem in wavelet analysis. In particular, it is a hard work to construct the frame wavelets with explicit analytic forms. For a given n × n real expansive matrix A, the frame-sets with respect to A are a family of sets in R^n. Based on the frame-sets, a class of high-dimensional frame wavelets with analytic forms are constructed, which can be non-bandlimited, or even compactly supported. As an application, the construction is illustrated by several examples, in which some new frame wavelets with compact supports are constructed. Moreover, since the main result of this paper is about general dilation matrices, in the examples we present a family of frame wavelets associated with some non-integer dilation matrices that is meaningful in computational geometry.     

Pointwise Fatou theorem for generalized harmonic functions — Normal boundary values

For continuous positive solutions of the equation withp>(n+1)/2) in a half-space ₊ ⁿ⁺¹ the criterion is proved for existence of finite or infinite normal limit values at a given boundary point.     

Hardy Spaces on ? n with Pointwise Variable Anisotropy

In this work we develop highly geometric Hardy spaces, for the full range 0<p≤1. These spaces are constructed over multi-level ellipsoid covers of ℝ ⁿ that are highly anisotropic in the sense that the ellipsoids can change shape rapidly from point to point and from level to level. This generalizes previous work on anisotropic Hardy spaces where the geometry of the space was ‘fixed’ over ℝ ⁿ and extends Hardy spaces over spaces of homogeneous type, where the theory holds for p values that are ‘close’ to 1.     

Quadrature and Schatz’s Pointwise Estimates for Finite Element Methods

We investigate numerical integration effects on weighted pointwise estimates. We prove that local weighted pointwise estimates will hold, modulo a higher order term and a negative-order norm, as long as we use an appropriate quadrature rule. To complete the analysis in an application, we also prove optimal negative-order norm estimates for a corner problem taking into account quadrature. Finally, we present an example to show that our result is sharp. AMS subject classification (2000) 65N15     

Pointwise estimates of solutions for the multi-dimensional bipolar Euler–Poisson system

In the paper, we consider a multi-dimensional bipolar hydrodynamic model from semiconductor devices and plasmas. This system takes the form of Euler–Poisson with electric field and frictional damping added to the momentum equations. By making a new analysis on Green’s functions for the Euler system with damping and the Euler–Poisson system with damping, we obtain the pointwise estimates of the solution for the multi-dimensions bipolar Euler–Poisson system. As a by-product, we extend decay rates of the densities \({\rho_i(i=1,2)}\) in the usual L²-norm to the L^p-norm with \({p\geq1}\) and the time-decay rates of the momentums m_i(i = 1,2) in the L²-norm to the L^p-norm with p > 1 and all of the decay rates here are optimal.     

On Pointwise Convergence for Schrödinger Operator in a Convex Domain

Journal of Fourier Analysis and Applications - In this paper, we prove that the maximal inequality $$begin{aligned} big Vert sup _{|t|&lt;1}|e^{itDelta _D}f(x,y)|big Vert...     

Pointwise Estimates for the Bergman Kernel of the Weighted Fock Space

We prove upper pointwise estimates for the Bergman kernel of the weighted Fock space of entire functions in L ²(e ^−2φ ) where φ is a subharmonic function with Δφ a doubling measure. We derive estimates for the canonical solution operator to the inhomogeneous Cauchy-Riemann equation and we characterize the compactness of this operator in terms of Δφ.     

A Characterization of Pointwise Approximation for Linear Combinations of Szász-type Operators

In this paper, we use Ditzian-Totik modulus to study the pointwise approximation for the linear combinations of Szasz-type operators, and obtain the direct and converse theorem and the characterization of the pointwise approximation for the linear combinations of Szasz-type operators.     

Traveling Wavelets for the Simulation of a Hele–Shaw Flow with Surface Tension

The singularity dynamics of two immiscible fluids in a Hele-Shaw cell is studied using the traveling wavelet method. The singularities of the conformal mapping of the flow are accurately approximated by the solutions of a system of ordinary differential equations for the position of couples of zeros and poles. This model also describes secondary tip-splitting instabilities and remains valid for values of the surface tension and time intervals where previous perturbative calculations do not apply.     

Pointwise Inequalities of Landau–Kolmogorov Type for Functions Defined on a Finite Segment

For arbitrary t [0, 1], s [1, ], and A 2, we determine the unimprovable constant B for the inequality

Problems on Pointwise Convergence of Solutions to the Schr?dinger Equation

In this paper we consider several variants of the pointwise convergence problem for the Schr?dinger equation, which generalize the previously known results.