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

提案编号：摘要小波展开是一种数学工具，可以同时在多个尺度和位置上分析函数和数据。研究人员首先试图刻画所有（非紧）小波框架-这些framespermit更多的灵活性比广泛使用的orthonormal小波。然后，他的目标是表征和找到例子的小波，其伸缩矩阵扩大在某些方向，但不是在其他。另一个目标是证明“墨西哥帽”小波族在所有勒贝格空间中都是稠密的，以便该族可用于在不仅仅是均方意义上近似数据。在另一个不同的主题，调查员也将追求的基本模型方程，薄流体膜运动的基础问题。所提出的研究旨在从数学上确定这些方程稳态的稳定性。特别是，“液滴”的稳定状态将进行研究。这种稳定的稳态的存在表明了在电影中创造一种模式的可能性。小波理论借鉴了基础数学和工程学科，如信号处理，以建立有效分析和存储信息的工具。这两个理论是：近年来，基础研究和实际应用之间的一条通道特别有效，从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基础研究。关于流体流动的数学方程的许多问题是出了名的难。鉴于这一困难，许多研究都集中在特殊情况下，例如位于平面上或悬挂在平面上的液体薄膜。这些薄膜出现在许多工业涂层的情况下，如照相胶片的制造，或磁盘驱动器的涂层。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，在一维空间中所知的也比在物理相关的二维空间中所知的要多得多，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.