The Nonexistence of Shearlet-Like Scaling Multifunctions that Satisfy Certain Minimally Desirable Properties and Characterizations of the Reproducing Properties of the Integer Lattice Translations of a Countable Collection of Square Integrable Functions
Date of Award
Doctor of Philosophy (PhD)
Chair and Committee
In Chapter 1, we introduce three varieties of reproducing systems—Bessel systems, frames, and Riesz bases—within the Hilbert space context and prove a number of elementary results, including qualitative characterizations of each and several results regarding the combination and partitioning of reproducing systems.
In Chapter 2, we characterize when the integer lattice translations of a countable collection of square integrable functions forms a Bessel system, a frame, and a Riesz basis.
In Chapter 3, we introduce composite wavelet systems and generalize several well-known classical wavelet system results—including those regarding pointwise values of the Fourier transform of the wavelet and scaling function and those regarding dependencies on the multiresolution analysis defining properties—to the composite case. Two corollaries of these results are the nonexistence of composite scaling multifunctions of Haar-type, when the composite dilation group is infinite, and the nonexistence of classical multiwavelets, when the dilation matrix is integral and has determinant 1 in absolute value.
There is a well-known connection, via the Fourier transform, between smoothness and integral polynomial decay. In Chapter 4, we prove several generalized versions of this result in which smoothness and integral polynomial decay are replaced with Hölder continuity and fractional polynomial decay; logarithmic continuity and logarithmic decay; iterated Hölder continuity and multivariable fractional polynomial decay.
In Chapter 5, we prove the nonexistence of shearlet-like scaling multifunctions that satisfy a minimal amount of decay and either a minimal amount of regularity or one of two “finite type” conditions.
In Chapter 6, we indicate a number of interesting questions that arise from the reproducing system characterizations of Chapter 2 and the scaling multifunction nonexistence results of Chapters 3 and 5.
Houska, Robert, "The Nonexistence of Shearlet-Like Scaling Multifunctions that Satisfy Certain Minimally Desirable Properties and Characterizations of the Reproducing Properties of the Integer Lattice Translations of a Countable Collection of Square Integrable Functions" (2009). All Theses and Dissertations (ETDs). 157.