Wavelet Frames and Bases, and Fourth Order "thin film" Eigenproblems

摘要小波扩展是一种数学工具，可以同时在多个尺度和位置分析函数和数据。研究者首先寻求表征所有（非紧）小波帧-这些帧允许更多的灵活性比广泛使用的标准正交小波。然后，他的目标是描述并找到膨胀矩阵在某些方向上扩展而在其他方向上不扩展的小波的例子。另一个目标是证明“墨西哥帽”小波族在所有Lebesgue空间中都是密集的，因此这个小波族可以用来逼近数据，而不仅仅是在均方意义上。在另一个不同的主题上，研究者还将探讨关于基本模型方程的问题，这些方程是薄流体薄膜运动的基础。提出的研究旨在数学上确定这些方程稳态的稳定性。特别是，“液滴”稳态将被研究。这种稳定的稳定状态的存在将预示着在电影中创造一种模式的可能性。小波理论利用基础数学和信号处理等工程学科来创建有效分析和存储信息的工具。近年来，基础研究和实际应用之间的双向通道特别有效。谐波分析的抽象数学理论已经转化为大规模的工程解决方案（例如，联邦调查局使用小波压缩技术来存储其指纹图像），而工程挑战继续刺激数学的基础研究。许多关于流体流动数学方程的问题都是出了名的困难。鉴于这一困难，许多研究集中在特殊情况下，如薄膜液体坐在或挂在平面上。这些胶片出现在许多工业涂布情况下，例如照相胶片的制造或磁盘驱动器的涂布。对这些问题的数学理解直到20世纪90年代才变得实质性，即使是现在，人们对一维空间的了解也比对二维空间的物理相关情况的了解要多得多，而二维空间是本研究的重点。

Proposal Number: DMS-0140481PI: Richard LaugesenABSTRACTWavelet expansions are a mathematical tool that enablefunctions and data to be analyzed at multiple scales andlocations simultaneously. The investigator seeks first tocharacterize all (non-tight) wavelet frames - these framespermit more flexibility than the widely-used orthonormalwavelets. Then he aims to characterize and find examplesof wavelets whose dilation matrices expand in some directionsbut not in others. Another goal is to prove that the"Mexican hat" wavelet family is dense in all Lebesgue spaces,so that this family can be used to approximate data in morethan just the mean-square sense. On a different topic, theinvestigator will also pursue questions about the fundamentalmodel equations that underlie motion of thin fluid films. Theproposed research aims to mathematically determine thestability of steady states of these equations. In particular,"droplet" steady states will be studied. The existence of suchstable steady states would signal the possibility of creatinga pattern in the film.Wavelet theory draws on fundamental mathematics and onengineering disciplines such as signal processing to createtools for efficiently analyzing and storing information.The two-way street between basic research and practicalapplications has been particularly effective in recent years.Abstract mathematical theories from harmonic analysis havebeen transformed into large scale engineering solutions (forexample the FBI uses a wavelet compression technique to storeits fingerprint images), while engineering challenges continueto stimulate fundamental research in mathematics. Manyquestions about the mathematical equations of fluid flow arefamously difficult. In view of this difficulty, much researchhas concentrated on special situations, such as a thin film offluid either sitting on or hanging from a flat surface. Thesefilms arise in many industrial coating situations, such as themanufacture of photographic film, or the coating of magneticdisk drives. The mathematical understanding of these problemsbecame substantial only in the 1990s, and even now, much moreis known in one space dimension than in the physically relevantcase of two space dimensions, where this research willconcentrate.