On the General Solution of the Ultrahyperbolic Bessel Operator

We study the general solution of equation □B,cku(x)=f(x), where □B,ck is the ultrahyperbolic Bessel operator iterated k-times and is defined by □B,ck=[(1/c2)(Bx1+Bx2+⋯+Bxp)−(Bxp+1+⋯+Bxp+q)],  p+q=n, n is the dimension of ℝn+={x:x=(x1,x2,…,xn),  x1>0,…,xn>0}, Bxi=∂2/∂xi2+(2vi/xi)(∂/∂xi), 2vi=2βi+1, βi>−1/2, xi>0 (i=1,2,…,n), f(x) is a given generalized function, u(x) is an unknown generalized function, k is a nonnegative integer, c is a positive constant, and x∈Rn+

Representation of Operators in the Time-Frequency Domain and Generalzed Gabor Multipliers

Starting from a general operator representation in the time-frequency domain, this paper addresses the problem of approximating linear operators by operators that are diagonal or band-diagonal with respect to Gabor frames. A haracterization of operators that can be realized as Gabor multipliers is given and necessary conditions for the existence of (Hilbert-Schmidt) optimal Gabor multiplier approximations are discussed and an efficient method for the calculation of an operator\u27s best approximation by a Gabor multiplier is derived. The spreading function of Gabor multipliers yields new error estimates for these approximations. Generalizations (multiple Gabor multipliers) are introduced for better approximation of overspread operators. The Riesz property of the projection operators involved in generalized Gabor multipliers is characterized, and a method for obtaining an operator\u27s best approximation by a multiple Gabor multiplier is suggested. Finally, it is shown that in certain situations, generalized Gabor multipliers reduce to a finite sum of regular Gabor multipliers with adapted windows

On the time-frequency representation of operators and generalized Gabor multiplier approximations

28 pagesInternational audienceStarting from a general operator representation in the time-frequency do- main, this paper addresses the problem of approximating linear operators by operators that are diagonal or band-diagonal with respect to Gabor frames. A characterization of operators that can be realized as Gabor multipliers is given and necessary conditions for the existence of (Hilbert-Schmidt) optimal Gabor multiplier approximations are discussed and an efficient method for the calculation of an operator's best approximation by a Gabor multiplier is derived. The spreading function of Gabor multipliers yields new error estimates for these approximations. Generalizations (multiple Gabor multipliers) are introduced for better approximation of overspread operators. The Riesz property of the projection operators involved in generalized Gabor multipliers is characterized, and a method for obtaining an operator's best approximation by a multiple Gabor multiplier is suggested. Finally, it is shown that in certain situations, generalized Gabor multipliers reduce to a finite sum of regular Gabor multipliers with adapted windows

Gabor frames and the fractional Fourier transform

Ziel der vorliegenden Diplomarbeit ist, die Auswirkung der fraktionalen Fourier Transformantion auf Gabor Frames mit Gaußscher Fensterfunktion zu untersuchen. Ausgehend von solch einem Gabor-System wird auf jedes Element eine Wurzel der Fourier Transformation angewandt. Die entstehenden Systeme kann man sich geometrisch als eine Familie rotierter Ellipsen vorstellen, die entlang eines Gitters in der Zeit-Frequenz Ebene positioniert sind. Es wird ersichtlich, wie sich die Anordnung dieser Ellipsen in den Frame-Schranken widerspiegelt. Nachdem in den Kapiteln 1 bis 3 die Grundlagen der Fourier- und Gabor Analyisis beschrieben werden, widmet sich Kapitel 4 der fraktionalen Fourier Transformation und ihrer Implementation. Hierzu wird ein etablierter Algorithmus (siehe [3]) mit einem in der NuHAG entwickelten Verfahren verglichen, für das auch eine Konvergenzanalyse durchgeführt wird. Kapitel 5 beschreibt die Ergebnisse numerischer Experimente zu den oben genannten Systemen. Es zeigt sich, dass falls das resultierende System ein Gabor Frame ist, das Verhalten der Frameschranken leicht verständlich, aber durchaus erstaunlich ist. Auÿerdem werden einige Spezialfälle betrachtet, sowie jene Systeme, die durch eine zufällige Wahl der Rotationsparameter entstehen. In Kapitel 6 schließlich finden sich Stabilitätsresultate für die Frame-Schranken der untersuchten Systeme. Diese werden mit Hilfe der Theorie der Wiener- Amalgam Räumen bewiesen

An annotated bibliography for comparative prime number theory

The goal of this annotated bibliography is to record every publication on the topic of comparative prime number theory together with a summary of its results. We use a unified system of notation for the quantities being studied and for the hypotheses under which results are obtained. We encourage feedback on this manuscript (see the end of Section~1 for details).Comment: 98 pages; supersedes "Comparative prime number theory: A survey" (arXiv:1202.3408

Fourier Transform

The application of Fourier transform (FT) in signal processing and physical sciences has increased in the past decades. Almost all the textbooks on signal processing or physics have a section devoted to the FT theory. For this reason, this book focuses on signal processing and physical sciences. The book chapters are related to fast hybrid recursive FT based on Jacket matrix, acquisition algorithm for global navigation satellite system, determining the sensitivity of output parameters based on FFT, convergence of integrals of products based on Riemann-Lebesgue Lemma function, extending the real and complex number fields for treating the FT, nonmaterial structure, Gabor transform, and chalcopyrite bioleaching. The book provides applications oriented to signal processing and physics written primarily for engineers, mathematicians, physicians and graduate students, will also find it useful as a reference for their research activities

Improvements In Interferometric Data Modeling For The New Era Of Radio Cosmology

The redshifted 21 cm line promises to provide a wealth of information about the evolution of our universe but remains as yet undetected. The general theme of this thesis is developing increasingly realistic models of the raw data we collect from a radio telescope. This is important because at the end of the day extracting the cosmological signal from the data will be accomplished by achieving a level of understanding of all the possible alternative sources that might mimic the cosmological signal, to a degree that we can confidently reject those alternatives as causes of our detection. The work presented in this thesis has been done in the context of working on the HERA experiment which aims to make the first measurements of spatial fluctuations in the emission from neutral hydrogen. In this thesis I emphasized aspects of the visibility function that are important for efficient and realistic visibility simulations including full account of polarization effects, in particular using a harmonic parameterization of the integrand. I assessed the effect of potential ionospheric attenuation on the suppression of polarization contamination in 21 cm power spectrum measurements using visibility simulations based on historical ionospheric plasma density data. I showed how we can use closed-form calculations of the cross-frequency angular power spectrum on the sky to generate simple mock cosmological signal simulations that are useful for validating data analysis methods. I showed how the window functions associated with a 21 cm power spectrum estimate can be approximated by simple forms that are much cheaper to evaluate than the general definition. Finally, I produced a new Southern Sky Model that combines the best available diffuse radio emission surveys that cover HERA\u27s field of view and observing bandwidth with a point source catalog without double counting flux

Errata and Addenda to Mathematical Constants

We humbly and briefly offer corrections and supplements to Mathematical Constants (2003) and Mathematical Constants II (2019), both published by Cambridge University Press. Comments are always welcome.Comment: 162 page