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

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

Proposal Number: DMS-0140481 PI: Richard Laugesen ABSTRACT Wavelet expansions are a mathematical tool that enable functions and data to be analyzed at multiple scales and locations simultaneously. The investigator seeks first to characterize all (non-tight) wavelet frames - these frames permit more flexibility than the widely-used orthonormal wavelets. Then he aims to characterize and find examples of wavelets whose dilation matrices expand in some directions but 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 more than just the mean-square sense. On a different topic, the investigator will also pursue questions about the fundamental model equations that underlie motion of thin fluid films. The proposed research aims to mathematically determine the stability of steady states of these equations. In particular, "droplet" steady states will be studied. The existence of such stable steady states would signal the possibility of creating a pattern in the film. Wavelet theory draws on fundamental mathematics and on engineering disciplines such as signal processing to create tools for efficiently analyzing and storing information. The two-way street between basic research and practical applications has been particularly effective in recent years. Abstract mathematical theories from harmonic analysis have been transformed into large scale engineering solutions (for example the FBI uses a wavelet compression technique to store its fingerprint images), while engineering challenges continue to stimulate fundamental research in mathematics. Many questions about the mathematical equations of fluid flow are famously difficult. In view of this difficulty, much research has concentrated on special situations, such as a thin film of fluid either sitting on or hanging from a flat surface. These films arise in many industrial coating situations, such as the manufacture of photographic film, or the coating of magnetic disk drives. The mathematical understanding of these problems became substantial only in the 1990s, and even now, much more is known in one space dimension than in the physically relevant case of two space dimensions, where this research will concentrate.