Adaptive non-uniform B-spline dictionaries on a compact interval

Cardinal B-spline dictionaries on a compact interval

A prescription for constructing dictionaries for cardinal spline spaces on a compact interval is provided. It is proved that such spaces can be spanned by dictionaries which are built by translating a prototype B-spline function of fixed support into the knots of the required cardinal spline space. This implies that cardinal spline spaces on a compact interval can be spanned by dictionaries of cardinal B-spline functions of broader support that the corresponding basis function.

From cardinal spline wavelet bases to highly coherent dictionaries

Wavelet families arise by scaling and translations of a prototype function, called the {\em {mother wavelet}}. The construction of wavelet bases for cardinal spline spaces is generally carried out within the multi-resolution analysis scheme. Thus, the usual way of increasing the dimension of the multi-resolution subspaces is by augmenting the scaling factor. We show here that, when working on a compact interval, the identical effect can be achieved without changing the wavelet scale but reducing the translation parameter. By such a procedure we generate a redundant frame, called a {\em{diction…

对称性、尺度与最优时频集中：分段多项式与小波字典中海森堡不确定性的极小化

Symmetry, Scaling, and Optimal Time-Frequency Concentration: Minimising the Heisenberg Uncertainty in Piecewise-Polynomial and Wavelet Dictionaries

本文提出函数类层级与定制不确定性算子，证明不确定性在尺度平移下不变，帐篷函数为两个小波字典中时频不确定性的唯一极小化器（U=3/10），且rect的p重自卷积不确定性随p增大单调趋向海森堡下界1/4，为自适应小波设计提供原则性方法。

Local and Global Phaseless Sampling in Real Spline Spaces

We study the recovery of functions in real spline spaces from unsigned sampled values. We consider two types of recovery. The one is to recover functions locally from finitely many unsigned samples. And the other is to recover functions on the whole line from infinitely many unsigned samples. In both cases, we give characterizations for a sequence of distinct points to be a phaseless sampling sequence, at which any nonseparable function is determined up to a sign on an interval or on the whole line by its unsigned sampled values. Moreover, for the case of local recovery, we also study the almo…

Deconvolution of band limited functions on non-compact symmetric spaces

It is shown that a band limited function on a non-compact symmetric space can be reconstructed in a stable way from some countable sets of values of its convolution with certain distributions of compact support. A reconstruction method in terms of frames is given which is a generalization of the classical result of Duffin-Schaeffer about exponential frames on intervals. The second reconstruction method is given in terms of polyharmonic average splines.

基于张量积的单位正方形上二元 Bessel 位势空间的 Quarklet 刻画

Quarklet Characterizations for bivariate Bessel-Potential Spaces on the Unit Square via Tensor Products

本文利用 B 样条 quarklet 及张量积结构，结合一元边界自适应 quarklet 与 Hansen 和 Sickel 的结果，推导出了单位正方形上二元 Bessel 位势空间的新刻画。