Compactly Supported One-cyclic Wavelets Derived from Beta Distributions

A Douglas-Rachford construction of non-separable continuous compactly supported multidimensional wavelets

After re-casting the $n$-dimensional wavelet construction problem as a feasibility problem with constraints arising from the requirements of compact support, smoothness and orthogonality, the Douglas--Rachford algorithm is employed in the search for one- and two-dimensional wavelets. New one-dimensional wavelets are produced as well as genuinely non-separable two-dimensional wavelets in the case where the dilation on the plane is the standard $D_af(t)=a^{-1}f(t/a)$ $(t\in{\mathbb R}^n, a>0)$.

Orthonormal Compactly Supported Wavelets with Optimal Sobolev Regularity

Numerical optimization is used to construct new orthonormal compactly supported wavelets with Sobolev regularity exponent as high as possible among those mother wavelets with a fixed support length and a fixed number of vanishing moments. The increased regularity is obtained by optimizing the locations of the roots the scaling filter has on the interval (pi/2,π). The results improve those obtained by I. Daubechies [Comm. Pure Appl. Math. 41 (1988), 909-996], H. Volkmer [SIAM J. Math. Anal. 26 (1995), 1075-1087], and P. G. Lemarie-Rieusset and E. Zahrouni [Appl. Comput. Harmon. Anal. 5 (1998), …

Multiwavelet density estimation

Accurate density estimation methodologies play an integral role in a variety of scientific disciplines, with applications including simulation models, decision support tools, and exploratory data analysis. In the past, histograms and kernel density estimators have been the predominant tools of choice, primarily due to their ease of use and mathematical simplicity. More recently, the use of wavelets for density estimation has gained in popularity due to their ability to approximate a large class of functions, including those with localized, abrupt variations. However, a well-known attribute of …

Asymptotic normality of simultaneous estimators of cyclic long-memory processes

Spectral singularities at non-zero frequencies play an important role in investigating cyclic or seasonal time series. The publication [2] introduced the generalized filtered method-of-moments approach to simultaneously estimate singularity location and long-memory parameters. This paper continues studies of these simultaneous estimators. A wide class of Gegenbauer-type semi-parametric models is considered. Asymptotic normality of several statistics of the cyclic and long-memory parameters is proved. New adjusted estimates are proposed and investigated. The theoretical findings are illustrated…

Sobolev空间中拟投影算子与对偶小波框架的逼近

Approximation by quasi-projection operators and dual wavelet frames in Sobolev spaces

研究Sobolev空间中拟投影算子为函数及其导数提供同时逼近和密度阶的条件，利用所得准则赋予对偶Sobolev空间中基于MRA的对偶小波框架所需同时逼近性质。

Stability properties for a compactly supported prescale function

We show that if $ϕ$ is a continuous, minimally supported prescale function, then its translates are linearly independent on any set of positive measure in the unit interval. This generalizes results of Y. Meyer and P. G. Lemarie. This result implies that a stability condition, introduced by Gundy and Kazarian for the study of local convergence of spline wavelet expansions, is satisfied for all expansions arizing from multiresolution analyses generated by such prescale functions $ϕ$.