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

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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.