Wavelet and frame theory: frame bound gaps, generalized shearlets, Grassmannian fusion frames, and p-adic wavelets

Abstract

The first wavelet system was discovered by Alfréd Haar one hundred years ago. Since then the field has grown enormously. In 1952, Richard Duffin and Albert Schaeffer synthesized the earlier ideas of a number of illustrious mathematicians into a unified theory, the theory of frames. Interest in frames as intriguing objects in their own right arose when wavelet theory began to surge in popularity. Wavelet and frame analysis is found in such diverse fields as data compression, pseudo-differential operator theory and applied statistics.

We shall explore five areas of frame and wavelet theory: frame bound gaps, smooth Parseval wavelet frames, generalized shearlets, Grassmannian fusion frames, and <italic>p</italic>-adic wavlets. The phenomenon of a <italic>frame bound gap</italic> occurs when certain sequences of functions, converging in <italic>L^2</italic> to a Parseval frame wavelet, generate systems with frame bounds that are uniformly bounded away from <italic>1</italic>. In the 90's, Bin Han proved the existence of Parseval wavelet frames which are smooth and compactly supported on the frequency domain and also approximate wavelet set wavelets. We discuss problems that arise when one attempts to generalize his results to higher dimensions.

A shearlet system is formed using certain classes of dilations over <italic>R^2</italic> that yield directional information about functions in addition to information about scale and position. We employ representations of the extended metaplectic group to create shearlet-like transforms in dimensions higher than <italic>2</italic>. Grassmannian frames are in some sense optimal representations of data which will be transmitted over a noisy channel that may lose some of the transmitted coefficients. Fusion frame theory is an incredibly new area that has potential to be applied to problems in distributed sensing and parallel processing. A novel construction of Grassmannian fusion frames shall be presented. Finally, <italic>p</italic>-adic analysis is a growing field, and <italic>p</italic>-adic wavelets are eigenfunctions of certain pseudo-differential operators. A construction of a <italic>p</italic>-adic wavelet basis using dilations that have not yet been used in <italic>p</italic>-adic analysis is given.