While the spectral theory of compact operators is known to many, knowledge regarding the relationship between eigenvalues and approximation numbers might be less known. By examining these numbers in tandem, one may develop a link between eigenvalues and l^p spaces. In this paper, we develop the background of this connection with in-depth examples.

The exponential nature of Moore's law has inadvertently created huge data storage complexes that are scattered around the world. Data elements are continuously being searched, processed, erased, combined and transferred to other storage units without much regard to power consumption. The need for faster searches and power efficient data processing is becoming a fundamental requirement. Quantum computing may offer an elegant solution to speed and power through the utilization of the natural laws of quantum mechanics. Therefore, minimal cost quantum circuit implementation methodologies are greatly desired. This thesis explores the decomposition of group functions and the Walsh spectrum for implementing quantum canonical cascades with minimal cost. Three different methodologies, using group decomposition, are presented and generalized to take advantage of different quantum computing hardware, such as ion traps and quantum dots. Quantum square root of swap gates and fixed angle rotation gates comprise the first two methodologies. The third and final methodology provides further quantum cost reduction by more efficiently utilizing Hilbert spaces through variable angle rotation gates. The thesis then extends the methodology to realize a robust quantum circuit synthesis tool for single and multi-output quantum logic functions.

A beamforming antenna array is a set of antennas whose outputs are weighted by complex values and combined to form the array output. The effect of the complex valued weights is to steer lobes and nulls of the array pattern to desired directions. These directions may be unknown and so the antenna weights must be adjusted adaptively until some measure of array performance is improved, indicating proper lobe or null placement. An adaptive algorithm to adjust the complex weights of an antenna array is presented that nulls high power signals while allowing reception of GPS signals as long as the signals arrive from different directions. The GPS signals are spread spectrum modulated and have very low average power, on the order of background thermal noise. Simulations of the adaptive algorithm minimize the output power of the array to within 5 dB of the background noise level. The adaptive algorithm, named the Hilbert-space-based (HSB) gradient method, is based on the steepest descent algorithm and implements an efficient, exact gradient calculation. With M antennas in the array, only M − 1 weights are adjustable; one antenna weight is held constant to prevent the algorithm from minimizing the output power trivially by zeroing all weights thus preventing the reception of any signal by the array. It appears that M − 1 adjustable antenna weights can null M − 1 unwanted signals (jammers). However, in the course of the algorithm development, a few configurations of antennas and jammer arrival directions were found where this is not true. Even when the jammer arrival directions are known (‘oracle’) certain configurations are mathematically impossible to cancel. The oracle solution has a matrix formulation and under certain conditions an exact solution for antenna weights to annihilate the jammers can be found. This provides an excellent comparison tool to assess the performance of other adaptive algorithms. The HSB gradient adaptive algorithm and the oracle solution are both implemented in Matlab. Outputs of both are plotted for comparison.

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Let (b$_{i,j}$) be a bounded matrix on extit{ l}$^{2}$, $Bbb T={zinBbb C:|z|=1}$, and A be a bounded matrix on L$^{ 2}(mathbb{T)}$ satisfying the conditions 1.$langle Az^{2j},z^{2i} angle =sigma ^{-1}b_{ij}+|alpha |^{2}sigma ^{-1}langle Az^{j},z^{i} angle $; 2.$langle Az^{2j},z^{2i-1} angle =-alpha sigma ^{-1}b_{ij}+alpha sigma ^{-1}langle Az^{j},z^{i} angle $; 3.$langle Az^{2j-1},z^{2i} angle =-overline{alpha }sigma ^{-1}b_{ij}+overline{alpha }sigma ^{-1}langle Az^{j},z^{i} angle$; 4.$langle Az^{2j-1},z^{2i-1} angle =|alpha |^{2}sigma ^{-1}b_{ij}+sigma ^{-1}langle Az^{j},z^{i} angle $ hspace{-0.76cm} for all $i,jin mathbb{Z}$, where $sigma =1+|alpha |^{2},,alpha in mathbb{C},alpha eq0$. The above conditions evidently suggests that there is a "dyadic" relation in the entries of $A$. Here in the following picture illustrates how each $ij-$th entry of $A$ generates the 2 by 2 block in $A$ with entries ${a_{2i 2j}, a_{2i-1 2j}, a_{2i 2j-1}, a_{2i-1 2j-1}}.$ vspace{-0.3cm} egin{figure}[hp] egin{center} includegraphics[scale=0.42]{cubic.pdf} end{center} vspace{-0.8cm}caption{The dyadic recurrent form} end{figure} It has been shown [2] that $displaystyle A=sum_{n=0}^{infty }S^{n}BS^{ast n}$, where $Sz^i=sigma ^{-1/2}(overline{alpha }z^{2i}+z^{2i-1})$ and $$B=sumlimits_{i=-infty}^infty sumlimits_{j=-infty}^infty b_{ij}(u_{i}otimes u_{j}), u_{i}(z)=sigma ^{-1/2}z^{2i-1}(alpha -z).$$ In this paper, we shall use the above relations to compute $langle a_{i,j} angle $ explicitly. ewline Key words: shift operator, bounded matrix, dyadic recurrent formula, slant Toeplitz operator, separable Hilbert space 2.$langle Az^{2j},z^{2i-1} angle =-alpha sigma ^{-1}b_{ij}+alpha sigma ^{-1}langle Az^{j},z^{i} angle $ 3.$langle Az^{2j-1},z^{2i} angle =-overline{alpha }sigma ^{-1}b_{ij}+overline{alpha }sigma ^{-1}langle Az^{j},z^{i} angle $ 4.$langle Az^{2j-1},z^{2i-1} angle =|alpha |^{2}sigma ^{-1}b_{ij}+sigma ^{-1}langle Az^{j},z^{i} angle $ for all $i,jin mathbb{Z}$, where $sigma =1+|alpha |^{2},,alpha in mathbb{C},alpha eq0$ egin{figure}[hp] egin{center} includegraphics[scale=0.42]{cubic.pdf} end{center} caption{The dyadic recurrent form} end{figure} Since it has been shown [2] that $displaystyle A=sum_{n=0}^{infty }S^{n}BS^{ast n}$, where $ Sz^i=sigma ^{-1/2}(overline{alpha }z^{2i}+z^{2i-1})$ $ B=sum sum b_{ij}(u_{i}otimes u_{j})$ ;;; which $u_{i}(z)=sigma ^{-1/2}z^{2i-1}(alpha -z)$ Then we can use it to compute $langle Az^{j},z^{i} angle $ explicity if A satisfies the previous condition. ewline Key words: shift operator, bounded matrix, dyadic recurrent formula, slant Toeplitz operator, separable Hilbert space

Light transport simulation aims at the numerical computation of the propagation of visible electromagnetic energy in macroscopic environments. In this thesis, we develop the foundations for a modern theory of light transport simulation, unveiling the geometric structure of the continuous theory and providing a formulation of computational techniques that furnishes remarkably efficacy with only local information. Utilizing recent results from various communities, we develop the physical and mathematical structure of light transport from Maxwell's equations by studying a lifted representation of electromagnetic theory on the cotangent bundle. At the short wavelength limit, this yields a Hamiltonian description on six-dimensional phase space, with the classical formulation over the space of "positions and directions" resulting from a reduction to the five-dimensional cosphere bundle. We establish the connection between light transport and geometrical optics by a non-canonical Legendre transform, and we derive classical concepts from radiometry, such as radiance and irradiance, by considering measurements of the light energy density. We also show that in idealized environments light transport is a Lie-Poisson system for the group of symplectic diffeomorphisms, unveiling a tantalizing similarity between light transport and fluid dynamics. Using Stone's theorem, we also derive a functional analytic description of light transport. This bridges the gap to existing formulations in the literature and naturally leads to computational questions. We then address one of the central challenges for light transport simulation in everyday environments with scattering surfaces: how are efficient computations possible when the light energy density can only be evaluated pointwise? Using biorthogonal and possibly overcomplete bases formed by reproducing kernel functions, we develop a comprehensive theory for computational techniques that are restricted to pointwise information, subsuming for example sampling theorems, interpolation formulas, quadrature rules, density estimation schemes, and Monte Carlo integration. The use of overcomplete representations makes us thereby robust to imperfect information, as is often unavoidable in practical applications, and numerical optimization of the sampling locations leads to close to optimal techniques, providing performance which considerably improves over the state of the art in the literature.

Light transport simulation aims at the numerical computation of the propagation of visible electromagnetic energy in macroscopic environments. In this thesis, we develop the foundations for a modern theory of light transport simulation, unveiling the geometric structure of the continuous theory and providing a formulation of computational techniques that furnishes remarkably efficacy with only local information. Utilizing recent results from various communities, we develop the physical and mathematical structure of light transport from Maxwell's equations by studying a lifted representation of electromagnetic theory on the cotangent bundle. At the short wavelength limit, this yields a Hamiltonian description on six-dimensional phase space, with the classical formulation over the space of "positions and directions" resulting from a reduction to the five-dimensional cosphere bundle. We establish the connection between light transport and geometrical optics by a non-canonical Legendre transform, and we derive classical concepts from radiometry, such as radiance and irradiance, by considering measurements of the light energy density. We also show that in idealized environments light transport is a Lie-Poisson system for the group of symplectic diffeomorphisms, unveiling a tantalizing similarity between light transport and fluid dynamics. Using Stone's theorem, we also derive a functional analytic description of light transport. This bridges the gap to existing formulations in the literature and naturally leads to computational questions. We then address one of the central challenges for light transport simulation in everyday environments with scattering surfaces: how are efficient computations possible when the light energy density can only be evaluated pointwise? Using biorthogonal and possibly overcomplete bases formed by reproducing kernel functions, we develop a comprehensive theory for computational techniques that are restricted to pointwise information, subsuming for example sampling theorems, interpolation formulas, quadrature rules, density estimation schemes, and Monte Carlo integration. The use of overcomplete representations makes us thereby robust to imperfect information, as is often unavoidable in practical applications, and numerical optimization of the sampling locations leads to close to optimal techniques, providing performance which considerably improves over the state of the art in the literature.

Modern data collection methods are now frequently returning observations that should be viewed as the result of digitized recording or sampling from stochastic processes rather than vectors of finite length. In spite of great demands, only a few classification methodologies for such data have been suggested and supporting theory is quite limited. The focus of this dissertation is on discrimination and classification in this infinite dimensional setting. The methodology and theory we develop are based on the abstract canonical correlation concept of Eubank and Hsing (2005), and motivated by the fact that Fisher's discriminant analysis method is intimately tied to canonical correlation analysis. Specifically, we have developed a theoretical framework for discrimination and classification of sample paths from stochastic processes through use of the Loeve-Parzen isomorphism that connects a second order process to the reproducing kernel Hilbert space generated by its covariance kernel. This approach provides a seamless transition between the finite and infinite dimensional settings and lends itself well to computation via smoothing and regularization. In addition, we have developed a new computational procedure and illustrated it with simulated data and Canadian weather data.

In this paper, we establish several strong convergence results of general approximate proximal algorithm and general Bregman-function-based approximate proximal algorithm for solving the generalized variational inequality problem with pseudomonotone multifunction.

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