Representations of Toeplitz-plus-Hankel matrices using trigonometric transformations with application to fast matrix-vector multiplication

AbstractRepresentations of real Toeplitz and Toeplitz-plus-Hankel matrices are presented that involve real trigonometric transformations (DCT, DST, DHT) and diagonal matrices. These representations can be used for fast matrix-vector multiplication. In particular, it is shown that the multiplication of an n × n Toeplitz-plus-Hankel matrix by a vector requires only 4 transformations of length n plus O(n) operations

Fredholm theory for Wiener-Hopf plus Hankel operators

Doutoramento em MatemáticaNa presente tese consideramos combinações algébricas de operadores de Wiener-Hopf e de Hankel com diferentes classes de símbolos de Fourier. Nomeadamente, foram considerados símbolos matriciais na classe de elementos quase periódicos, semi-quase periódicos, quase periódicos por troços e certas funções matriciais sectoriais. Adicionalmente, foi dedicada atenção também aos operadores de Toeplitz mais Hankel com símbolos quase periódicos por troços e com símbolos escalares possuindo n pontos de discontinuidades quase periódicas usuais. Em toda a tese, um objectivo principal teve a ver com a obtenção de descrições para propriedades de Fredholm para estas classes de operadores. De forma a deduzir a invertibilidade lateral ou bi-lateral para operadores de Wiener-Hopf mais Hankel com símbolos matriciais AP foi introduzida a noção de factorização assimétrica AP. Neste âmbito, foram dadas condições suficientes para a invertibilidade lateral e bi-lateral de operadores de Wiener- Hopf mais Hankel com símbolos matriciais AP. Para tais operadores, foram ainda exibidos inversos generalizados para todos os casos possíveis. Para os operadores de Wiener-Hopf-Hankel com símbolos matriciais SAP e PAP foi deduzida a propriedade de Fredholm e uma fórmula para a soma dos índices de Fredholm destes operadores de Wiener-Hopf mais Hankel e operadores de Wiener-Hopf menos Hankel. Uma versão mais forte destes resultados foi obtida usando a factorização generalizada AP à direita. Foram analisados os operadores de Wiener-Hopf-Hankel com símbolos que apresentam determinadas propriedades pares e também com símbolos de Fourier que contêm matrizes sectoriais. Em adição, para operadores de Wiener-Hopf-Hankel, foi obtido um resultado correspondente ao teorema clássico de Douglas e Sarason conhecido para operadores de Toeplitz com símbolos sectoriais e unitários. Condições necessárias e suficientes foram também deduzidas para que os operadores de Wiener-Hopf mais Hankel com símbolos L∞ sejam de Fredholm (ou invertíveis). Para se obter tal resultado, trabalhou-se com certas factorizações ímpares dos símbolos de Fourier. Os operadores de Toeplitz mais Hankel gerados por símbolos que possuem n pontos de discontinuidades quase periódicas usuais foram também considerados. Foram obtidas condições sob as quais estes operadores são invertíveis à direita e com dimensão de núcleo infinita, invertíveis à esquerda e com dimensão de co-núcleo infinita ou não normalmente solúveis. A nossa atenção foi também colocada em operadores de Toeplitz mais Hankel com símbolos matriciais contínuos por troços. Para tais operadores, condições necessárias e suficientes foram obtidas para se ter a propriedade de Fredholm. Tal foi realizado usando a abordagem do cálculo simbólico, determinados operadores auxiliares emparelhados com símbolos semi-quase periódicos e várias relações de equivalência após extensão entre operadores.In this thesis we considered algebraic combinations of Wiener-Hopf and Hankel operators with different classes of Fourier symbols. Namely, matrix symbols from the almost periodic, semi-almost periodic, piecewise almost periodic and certain sectorial matrix functions were considered. In addition, attention was also paid to Toeplitz plus Hankel operators with piecewise almost periodic symbols and with scalar symbols having n points of standard almost periodic discontinuities. In the entire thesis a main goal is to obtain Fredholm properties description of those classes of operators. To deduce the lateral or both sided invertibility theory for Wiener-Hopf plus Hankel operators with AP matrix symbols was introduced the notion of an AP asymmetric factorization. In this framework were given sufficient conditions for the lateral and both sided invertibility of the Wiener-Hopf plus Hankel operators with matrix AP symbols. For such kind of operators were also exhibited generalized inverses for all the possible cases. For the Wiener-Hopf-Hankel operators with matrix SAP and PAP symbols the Fredholm property and a formula for the sum of the Fredholm indices of these Wiener-Hopf plus Hankel and Wiener-Hopf minus Hankel operators were derived. A stronger version of these results was obtained by using the generalized right AP factorization. It was analyzed the Wiener-Hopf-Hankel operators with symbols presenting some even properties, and also with Fourier symbols which contain sectorial matrices. In addition, for Wiener-Hopf-Hankel operators, it was obtained a corresponding result to the classical theorem by Douglas and Sarason known for Toeplitz operators with sectorial and unitary valued symbols. Necessary and sufficient condition for the Wiener-Hopf plus Hankel operators with L∞ symbols to be Fredholm (or invertible) were also derived. To obtain such a result we dealt with certain odd asymmetric factorization of the Fourier symbols. The Toeplitz plus Hankel operators generated by symbols which have n points of standard almost periodic discontinuities were also considered. Conditions were obtained under which these operators are right-invertible and with infinite kernel dimension, left-invertible and with infinite cokernel dimension or simply not normally solvable. We also focused our attention to Toeplitz plus Hankel operators with piecewise almost periodic matrix symbols. For such operators necessary and sufficient conditions were obtained to have the Fredholm property. This was done using a symbol calculus approach, certain auxiliary paired operators with semi-almost periodic symbols, and several equivalence after extension operator relations

Monodromy dependence and connection formulae for isomonodromic tau functions

We discuss an extension of the Jimbo-Miwa-Ueno differential 1-form to a form closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational coefficients. This extension is based on the results of M. Bertola generalizing a previous construction by B. Malgrange. We show how this 1-form can be used to solve a long-standing problem of evaluation of the connection formulae for the isomonodromic tau functions which would include an explicit computation of the relevant constant factors. We explain how this scheme works for Fuchsian systems and, in particular, calculate the connection constant for generic Painlev\'e VI tau function. The result proves the conjectural formula for this constant proposed in \cite{ILT13}. We also apply the method to non-Fuchsian systems and evaluate constant factors in the asymptotics of Painlev\'e II tau function.Comment: 54 pages, 6 figures; v4: rewritten Introduction and Subsection 3.3, added few refs to match published articl

Gratings: Theory and Numeric Applications, Second Revisited Edition

International audienceThe second Edition of the Book contains 13 chapters, written by an international team of specialist in electromagnetic theory, numerical methods for modelling of light diffraction by periodic structures having one-, two-, or three-dimensional periodicity, and aiming numerous applications in many classical domains like optical engineering, spectroscopy, and optical telecommunications, together with newly born fields such as photonics, plasmonics, photovoltaics, metamaterials studies, cloaking, negative refraction, and super-lensing. Each chapter presents in detail a specific theoretical method aiming to a direct numerical application by university and industrial researchers and engineers.In comparison with the First Edition, we have added two more chapters (ch.12 and ch.13), and revised four other chapters (ch.6, ch.7, ch.10, and ch.11