On Gibbs constant for the Shannon wavelet expansion

Even though the Shannon wavelet is a prototype of wavelets, it lacks condition on decay which most wavelets are assumed to have. By providing a sufficient condition to compute the size of Gibbs phenomenon for the Shannon wavelet series, we can see the overshoot is propotional to the jump at discontinuity. By comparing it with that of the Fourier series, we also see that these two have exactly the same Gibbs constant.     

A summability for Meyer wavelets

The Gibbs’ phenomenon in the classical Fourier series is well-known. It is closely related with the kernel of the partial sum of the series. In fact, the Dirichlet kernel of the Fourier series is not positive. The poisson kernel of Cesaro summability is positive. As the consequence of the positiveness, the partial sum of Cesaro summability does not exhibit the Gibbs’ phenomenon. Most kernels associated with wavelet expansions are not positive. So wavelet series is not free from the Gibbs’ phenomenon. Because of the excessive oscillation of wavelets, we can not follow the techniques of the Fourier series to get rid of the unwanted quirk. Here we make a positive kernel for Meyer wavelets and as the result the associated summability method does not exhibit Gibbs’ phenomenon for the corresponding series.     

三角域上一类正交函数系的构造

V系统是作者2005年构造的一类L2[0,1]空间上的正交完备函数系. κ次V系统由κ次分片多项式组成,具有多分辨特性,是Haar小波函数的推广.基于V系统的正交表达,可以对CAGD中常见的几何模型用有限项V-级数做到精确重构,完全消除Gibbs现象,这是有限项Fourier级数或连续小波级数不能做到的.针对多变量情形,给出了三角域上的κ次正交V系统的构造方法.三角域上的V系统的重要应用显现在对3D复杂几何图组的整体频谱分析上.     

Orthogonal and Nonorthogonal Multiresolution Analysis, Scale Discrete and Exact Fully Discrete Wavelet Transform on the Sphere

Based on a new definition of dilation a scale discrete version of spherical multiresolution is described, starting from a scale discrete wavelet transform on the sphere. Depending on the type of application, different families of wavelets are chosen. In particular, spherical Shannon wavelets are constructed that form an orthogonal multiresolution analysis. Finally fully discrete wavelet approximation is discussed in the case of band-limited wavelets. June 18, 1996. Date revised: January 14, 1997.     

Detection of Singularities by Discrete Multiscale Directional Representations

One of the most remarkable properties of the continuous curvelet and shearlet transforms is their sensitivity to the directional regularity of functions and distributions. As a consequence of this property, these transforms can be used to characterize the geometry of edge singularities of functions and distributions by their asymptotic decay at fine scales. This ability is a major extension of the conventional continuous wavelet transform which can only describe pointwise regularity properties. However, while in the case of wavelets it is relatively easy to relate the asymptotic properties of the continuous transform to properties of discrete wavelet coefficients, this problem is surprisingly challenging in the case of discrete curvelets and shearlets where one wants to handle also the geometry of the singularity. No result for the discrete case was known so far. In this paper, we derive non-asymptotic estimates showing that discrete shearlet coefficients can detect, in a precise sense, the location and orientation of curvilinear edges. We discuss connections and implications of this result to sparse approximations and other applications.     

Basic properties of wavelets

A wavelet multiplier is a function whose product with the Fourier transform of a wavelet is the Fourier transform of a wavelet. We characterize the wavelet multipliers, as well as the scaling function multipliers and low pass filter multipliers. We then prove that if the set of all wavelet multipliers acts on the set of all MRA wavelets, the orbits are the sets of all MRA wavelets whose Fourier transforms have equal absolute values, and these are also equal to the sets, of all MRA wavelets with the corresponding scaling functions having the same absolute values of their Fourier transforms. As an application of these techniques, we prove that the set of MRA wavelets is arcwise connected in L²(R). Dedicated to Eugene Fabes The Wutam Consortium     

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.     

On wavelet sets

It is proved that associated with every wavelet set is a closely related “regularized” wavelet set which has very nice properties. Then it is shown that for many (and perhaps all) pairs E, F, of wavelet sets, the corresponding MSF wavelets can be connected by a continuous path in L²(ℝ) of MSF wavelets for which the Fourier transform has support contained in E ∪ F. Our technique applies, in particular, to the Shannon and Journe wavelet sets.     

Orthogonal Helmholtz decomposition in arbitrary dimension using divergence-free and curl-free wavelets

We present tensor-product divergence-free and curl-free wavelets, and define associated projectors. These projectors enable the construction of an iterative algorithm to compute the Helmholtz decomposition of any vector field, in wavelet domain. This decomposition is localized in space, in contrast to the Helmholtz decomposition calculated by Fourier transform. Then we prove the convergence of the algorithm in dimension two for any kind of wavelets, and in larger dimension for the particular case of Shannon wavelets. We also present a modification of the algorithm by using quasi-isotropic divergence-free and curl-free wavelets. Finally, numerical tests show the validity of this approach for a large class of wavelets.     

基于小波分析的空间机械臂运动规划的最优控制

讨论了空间机械臂系统非完整运动规划的最优控制问题.利用小波分析方法,将离散正交小波函数引入最优控制,由小波级数展开式逼近替代传统的Fourier基函数,提出基于小波分析的最优控制数值算法.仿真结果表明,该方法对求解空间机械臂非完整运动规划问题是有效的.     

Two-dimensional quaternion wavelet transform

In this paper we introduce the continuous quaternion wavelet transform (CQWT). We express the admissibility condition in terms of the (right-sided) quaternion Fourier transform. We show that its fundamental properties, such as inner product, norm relation, and inversion formula, can be established whenever the quaternion wavelets satisfy a particular admissibility condition. We present several examples of the CQWT. As an application we derive a Heisenberg type uncertainty principle for these extended wavelets.     

A Consistent and Stable Approach to Generalized Sampling

We consider the problem of generalized sampling, in which one seeks to obtain reconstructions in arbitrary finite dimensional spaces from a finite number of samples taken with respect to an arbitrary orthonormal basis. Typical approaches to this problem consider solutions obtained via the consistent reconstruction technique or as solutions of an overcomplete linear systems. However, the consistent reconstruction technique is known to be non-convergent and ill-conditioned in important cases, such as the recovery of wavelet coefficients from Fourier samples, and whilst the latter approach presents solutions which are convergent and well-conditioned when the system is sufficiently overcomplete, the solution becomes inconsistent with the original measurements. In this paper, we consider generalized sampling via a non-linear minimization problem and prove that the minimizers present solutions which are convergent, stable and consistent with the original measurements. We also provide analysis in the case of recovering wavelets coefficients from Fourier samples. We show that for compactly supported wavelets of sufficient smoothness, there is a linear relationship between the number of wavelet coefficients which can be accurately recovered and the number of Fourier samples available.     

Operator-Like Wavelet Bases of L_{2}(\mathbb{R}^{d})

The connection between derivative operators and wavelets is well known. Here we generalize the concept by constructing multiresolution approximations and wavelet basis functions that act like Fourier multiplier operators. This construction follows from a stochastic model: signals are tempered distributions such that the application of a whitening (differential) operator results in a realization of a sparse white noise. Using wavelets constructed from these operators, the sparsity of the white noise can be inherited by the wavelet coefficients. In this paper, we specify such wavelets in full generality and determine their properties in terms of the underlying operator.     

On wavelets interpolated from a pair of wavelet sets

We show that any wavelet, with the support of its Fourier transform small enough, can be interpolated from a pair of wavelet sets. In particular, the support of the Fourier transform of such wavelets must contain a wavelet set, answering a special case of an open problem of Larson. The interpolation procedure, which was introduced by X. Dai and D. Larson, allows us also to prove the extension property.

     

On multiresolution analysis (MRA) wavelets in ℝ n

We prove that for any expansive n×n integral matrix A with |det A|=2, there exist A-dilation minimally supported frequency (MSF) wavelets that are associated with a multiresolution analysis (MRA). The condition |det A|=2 was known to be necessary, and we prove that it is sufficient. A wavelet set is the support set of the Fourier transform of an MSF wavelet. We give some concrete examples of MRA wavelet sets in the plane. The same technique of proof is also applied to yield an existence result for A-dilation MRA subspace wavelets.     

多进小波展开的Gibbs函数的渐近性质

本文研究了一类多进小波展开的Gibbs函数的渐近性质，特别地，包括了多进Daubechies型小波．     

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.     

Average densities of frequency bands of wavelets

A function is called a wavelet if its integral translations and dyadic dilations form an orthonormal basis for L ²(?). The support of the Fourier transform of a wavelet is called its frequency band. In this paper, we study the relation between diameters and measures of frequency bands of wavelets, precisely say, we study the ratio of the measure to the diameter. This reflects the average density of the frequency band of a wavelet. In particular, for multiresolution analysis (MRA) wavelets, we do further research. First, we discuss the relation between diameters and measures of frequency bands of scaling functions. Next, we discuss the relation between frequency bands of wavelets and the corresponding scaling functions. Finally, we give the precise estimate of the measure of frequency bands of wavelets. At the same time, we find that when the diameters of frequency bands tend to infinity, the average densities tend to zero.     

Ripple marks in alwar quartzites east of mera ka gurha near udaipur,rajasthan

The present note includes a brief account of the ripple marks in the Alwar quartzites of an area East of the village Mera Ka Gurha near Udaipur. Both symmetrical and asymmetrical types of ripple marks have been noticed. In the case of asymmetrical ripples, the ripple index and the horizontal form index have been dealt with in detail. An attempt is made to decipher the palaeogeographical, ecological and other conditions that were present at the time of formation of the ripples.