We solve a one-sided Cauchy problem with zero right hand side modulo smooth errors for the wave operator associated to a smooth symmetric 2-tensor which is Lorentz on the interior and degenerate at the boundary. The degeneracy of the metric at the boundary gives rise to singularities in the wave operator. The initial data prescribed at the boundary must be modified from the classical Cauchy problem to suit the problem at hand. The problem is posed on the interior and the local solution is constructed using microlocal analysis and the techniques of Fourier Integral Operators. / Mathematics

This thesis addresses four topics in the area of applied harmonic analysis. First, we show that the affine densities of separable wavelet frames affect the frame properties. In particular, we describe a new relationship between the affine densities, frame bounds and weighted admissibility constants of the mother wavelets of pairs of separable wavelet frames. This result is also extended to wavelet frame sequences. Second, we consider affine pseudodifferential operators, generalizations of pseudodifferential operators that model wideband wireless communication channels. We find two classes of Banach spaces, characterized by wavelet and ridgelet transforms, so that inclusion of the kernel and symbol in appropriate spaces ensures the operator is Schatten p-class. Third, we examine the Schatten class properties of pseudodifferential operators. Using Gabor frame techniques, we show that if the kernel of a pseudodifferential operator lies in a particular mixed modulation space, then the operator is Schatten p-class. This result improves existing theorems and is sharp in the sense that larger mixed modulation spaces yield operators that are not Schatten class. The implications of this result for the Kohn-Nirenberg symbol of a pseudodifferential operator are also described. Lastly, Fourier integral operators are analyzed with Gabor frame techniques. We show that, given a certain smoothness in the phase function of a Fourier integral operator, the inclusion of the symbol in appropriate mixed modulation spaces is sufficient to guarantee that the operator is Schatten p-class.

Thesis (Ph. D.)--West Virginia University, 1998. / Title from document title page. Document formatted into pages; contains xi, 116 p. : ill. (some col.) Includes abstract. Includes bibliographical references (p. 94-98).

Fourier telescopy imaging is a recently-developed imaging method that relies on active structured-light illumination of the object. Reflected/scattered light is measured by a large “light bucket” detector; processing of the detected signal yields the magnitude and phase of spatial frequency components of the object reflectance or transmittance function. An inverse Fourier transform results in the image. In 2012 a novel method, known as time-average Fourier telescopy (TAFT), was introduced by William T. Rhodes as a means for diffraction-limited imaging through ground-level atmospheric turbulence. This method, which can be applied to long horizontal-path terrestrial imaging, addresses a need that is not solved by the adaptive optics methods being used in astronomical imaging. Field-experiment verification of the TAFT concept requires instrumentation that is not available at Florida Atlantic University. The objective of this doctoral research program is thus to demonstrate, in the absence of full-scale experimentation, the feasibility of time-average Fourier telescopy through (a) the design, construction, and testing of smallscale laboratory instrumentation capable of exploring basic Fourier telescopy datagathering operations, and (b) the development of MATLAB-based software capable of demonstrating the effect of kilometer-scale passage of laser beams through ground-level turbulence in a numerical simulation of TAFT. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2015. / FAU Electronic Theses and Dissertations Collection

Time Average Fourier Telescopy (TAFT) has been proposed as a means for obtaining high-resolution, diffraction-limited images over large distances through ground-level horizontal-path atmospheric turbulence. Image data is collected in the spatial-frequency, or Fourier, domain by means of Fourier Telescopy; an inverse two dimensional Fourier transform yields the actual image. TAFT requires active illumination of the distant object by moving interference fringe patterns. Light reflected from the object is collected by a “light-bucket” detector, and the resulting electrical signal is digitized and subjected to a series of signal processing operations, including an all-critical averaging of the amplitude and phase of a number of narrow-band signals. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2014. / FAU Electronic Theses and Dissertations Collection

In this dissertation we study non-negative self-adjoint Laplace type operators acting on smooth sections of a vector bundle. First, we assume base manifolds are compact, boundaryless, and Riemannian. We start from the Fourier integral operator representation of half-wave operators, continue with spectral zeta functions, heat and resolvent trace asymptotic expansions, and end with the quantitative Wodzicki residue method. In particular, all of the asymptotic coefficients of the microlocalized spectral counting function can be explicitly given and clearly interpreted. With the auxiliary pseudo-differential operators ranging all smooth endomorphisms of the given bundle, we obtain certain asymptotic estimates about the integral kernel of heat operators. As applications, we study spectral asymptotics of Dirac type operators such as characterizing those for which the second coefficient vanishes. Next, we assume vector bundles are trivial and base manifolds are Euclidean domains, and study non-negative self-adjoint extensions of the Laplace operator which acts component-wise on compactly supported smooth functions. Using finite propagation speed estimates for wave equations and explicit Fourier Tauberian theorems obtained by Yuri Safarov, we establish the principle of not feeling the boundary estimates for the heat kernel of these operators. In particular, the implied constants are independent of self-adjoint extensions. As a by-product, we affirmatively answer a question about upper estimate for the Neumann heat kernel. Finally, we study some specific values of the spectral zeta function of two-dimensional Dirichlet Laplacians such as spectral determinant and Casimir energy. For numerical purposes we substantially improve the short-time Dirichlet heat trace asymptotics for polygons. This could be used to measure the spectral determinant and Casimir energy of polygons whenever the first several hundred or one thousand Dirichlet eigenvalues are known with high precision by other means.