Bellman functions and Wavelets

Bellman函数法是一种功能强大的方法，可以用来约束由二进区间索引的和。近年来，用这种方法得到了许多谐波分析中的离散结果。迄今为止，它可以用于haar小波系统的标度函数或小波函数与函数内积的和，但不能用于一般小波系统的标度函数或小波函数。如果贝尔曼函数方法适用于更一般的小波，那么用目前的贝尔曼函数方法得到的许多离散结果可以扩展到连续设置。PI提出将Bellman函数的方法从目前的范围扩展到涉及任何MRA小波的和。小波在数学、物理、工程和信号处理等许多领域都很有用。它们是新的万维网图像压缩标准JPEG2000的基础。它们在谐波分析中也是一个有用的工具。经常需要估计涉及小波的和的大小。Bellman函数方法是一个非常有用的工具，但目前只适用于一种非常特殊的小波，称为Haar小波。PI有兴趣在可能的情况下将该方法推广到更一般的小波类。

Proposal Number: DMS-0140193PI: Janine WittwerABSTRACTThe method of Bellman functions is a powerful method thatcan be used to bound sums indexed by dyadic intervals.Many discrete results in Harmonic Analysis have beenobtained with this method in the last few years. To dateit can be used on sums involving inner products of afunction with the scaling or wavelet functions from theHaar wavelet system, but not the scaling or waveletfunctions of general wavelet systems. Many of the discreteresults obtained with the current Bellman function methodcould be expanded to the continuous setting if the Bellmanfunction method was available for more general wavelets.The PI proposes to expand the method of Bellman functionsfrom its current scope to sums involving any MRA wavelet.Wavelets have been useful in many areas such asmathematics, physics, engineering and signal processing.They are the basis of the new world wide web imagecompression standard JPEG2000. They prove to also be auseful tool in harmonic analysis. It is frequentlynecessary to estimate the size of sums involving wavelets.The Bellman function method is a very useful tool in doingso, but currently only works for a very special waveletcalled the Haar wavelet. The PI is interested ingeneralizing the method where possible to more generalclasses of wavelets.