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Anisotropic Hardy Spaces and Wavelets by Marcin Bownik Pdf
Investigates the anisotropic Hardy spaces associated with very general discrete groups of dilations. This book includes the classical isotropic Hardy space theory of Fefferman and Stein and parabolic Hardy space theory of Calderon and Torchinsky.
Pointwise Variable Anisotropic Function Spaces on Rn by Shai Dekel Pdf
Spaces of homogeneous type were introduced as a generalization to the Euclidean space and serve as a sufficient setting in which one can generalize the classical isotropic Harmonic analysis and function space theory. This setting is sometimes too general, and the theory is limited. Here, we present a set of flexible ellipsoid covers of Rn that replace the Euclidean balls and support a generalization of the theory with fewer limitations.
Convergence and Summability of Fourier Transforms and Hardy Spaces by Ferenc Weisz Pdf
This book investigates the convergence and summability of both one-dimensional and multi-dimensional Fourier transforms, as well as the theory of Hardy spaces. To do so, it studies a general summability method known as theta-summation, which encompasses all the well-known summability methods, such as the Fejér, Riesz, Weierstrass, Abel, Picard, Bessel and Rogosinski summations. Following on the classic books by Bary (1964) and Zygmund (1968), this is the first book that considers strong summability introduced by current methodology. A further unique aspect is that the Lebesgue points are also studied in the theory of multi-dimensional summability. In addition to classical results, results from the past 20-30 years – normally only found in scattered research papers – are also gathered and discussed, offering readers a convenient “one-stop” source to support their work. As such, the book will be useful for researchers, graduate and postgraduate students alike.
Real-Variable Theory of Hardy Spaces Associated with Generalized Herz Spaces of Rafeiro and Samko by Yinqin Li,Dachun Yang,Long Huang Pdf
The real-variable theory of function spaces has always been at the core of harmonic analysis. In particular, the real-variable theory of the Hardy space is a fundamental tool of harmonic analysis, with applications and connections to complex analysis, partial differential equations, and functional analysis. This book is devoted to exploring properties of generalized Herz spaces and establishing a complete real-variable theory of Hardy spaces associated with local and global generalized Herz spaces via a totally fresh perspective. This means that the authors view these generalized Herz spaces as special cases of ball quasi-Banach function spaces. In this book, the authors first give some basic properties of generalized Herz spaces and obtain the boundedness and the compactness characterizations of commutators on them. Then the authors introduce the associated Herz–Hardy spaces, localized Herz–Hardy spaces, and weak Herz–Hardy spaces, and develop a complete real-variable theory of these Herz–Hardy spaces, including their various maximal function, atomic, molecular as well as various Littlewood–Paley function characterizations. As applications, the authors establish the boundedness of some important operators arising from harmonic analysis on these Herz–Hardy spaces. Finally, the inhomogeneous Herz–Hardy spaces and their complete real-variable theory are also investigated. With the fresh perspective and the improved conclusions on the real-variable theory of Hardy spaces associated with ball quasi-Banach function spaces, all the obtained results of this book are new and their related exponents are sharp. This book will be appealing to researchers and graduate students who are interested in function spaces and their applications.
Hardy Spaces on Ahlfors-Regular Quasi Metric Spaces by Ryan Alvarado,Marius Mitrea Pdf
Systematically constructing an optimal theory, this monograph develops and explores several approaches to Hardy spaces in the setting of Alhlfors-regular quasi-metric spaces. The text is divided into two main parts, with the first part providing atomic, molecular, and grand maximal function characterizations of Hardy spaces and formulates sharp versions of basic analytical tools for quasi-metric spaces, such as a Lebesgue differentiation theorem with minimal demands on the underlying measure, a maximally smooth approximation to the identity and a Calderon-Zygmund decomposition for distributions. These results are of independent interest. The second part establishes very general criteria guaranteeing that a linear operator acts continuously from a Hardy space into a topological vector space, emphasizing the role of the action of the operator on atoms. Applications include the solvability of the Dirichlet problem for elliptic systems in the upper-half space with boundary data from Hardy spaces. The tools established in the first part are then used to develop a sharp theory of Besov and Triebel-Lizorkin spaces in Ahlfors-regular quasi-metric spaces. The monograph is largely self-contained and is intended for mathematicians, graduate students and professionals with a mathematical background who are interested in the interplay between analysis and geometry.
New Trends in Applied Harmonic Analysis, Volume 2 by Akram Aldroubi,Carlos Cabrelli,Stéphane Jaffard,Ursula Molter Pdf
This contributed volume collects papers based on courses and talks given at the 2017 CIMPA school Harmonic Analysis, Geometric Measure Theory and Applications, which took place at the University of Buenos Aires in August 2017. These articles highlight recent breakthroughs in both harmonic analysis and geometric measure theory, particularly focusing on their impact on image and signal processing. The wide range of expertise present in these articles will help readers contextualize how these breakthroughs have been instrumental in resolving deep theoretical problems. Some topics covered include: Gabor frames Falconer distance problem Hausdorff dimension Sparse inequalities Fractional Brownian motion Fourier analysis in geometric measure theory This volume is ideal for applied and pure mathematicians interested in the areas of image and signal processing. Electrical engineers and statisticians studying these fields will also find this to be a valuable resource.
Lebesgue Points and Summability of Higher Dimensional Fourier Series by Ferenc Weisz Pdf
This monograph presents the summability of higher dimensional Fourier series, and generalizes the concept of Lebesgue points. Focusing on Fejér and Cesàro summability, as well as theta-summation, readers will become more familiar with a wide variety of summability methods. Within the theory of higher dimensional summability of Fourier series, the book also provides a much-needed simple proof of Lebesgue’s theorem, filling a gap in the literature. Recent results and real-world applications are highlighted as well, making this a timely resource. The book is structured into four chapters, prioritizing clarity throughout. Chapter One covers basic results from the one-dimensional Fourier series, and offers a clear proof of the Lebesgue theorem. In Chapter Two, convergence and boundedness results for the lq-summability are presented. The restricted and unrestricted rectangular summability are provided in Chapter Three, as well as the sufficient and necessary condition for the norm convergence of the rectangular theta-means. Chapter Four then introduces six types of Lebesgue points for higher dimensional functions. Lebesgue Points and Summability of Higher Dimensional Fourier Series will appeal to researchers working in mathematical analysis, particularly those interested in Fourier and harmonic analysis. Researchers in applied fields will also find this useful.
This volume presents the recent theory of function spaces, paying special attention to some recent developments related to neighboring areas such as numerics, signal processing, and fractal analysis. Local building blocks, in particular (non-smooth) atoms, quarks, wavelet bases and wavelet frames are considered in detail and applied to diverse problems, including a local smoothness theory, spaces on Lipschitz domains, and fractal analysis.
Real-Variable Theory of Musielak-Orlicz Hardy Spaces by Dachun Yang,Yiyu Liang,Luong Dang Ky Pdf
The main purpose of this book is to give a detailed and complete survey of recent progress related to the real-variable theory of Musielak–Orlicz Hardy-type function spaces, and to lay the foundations for further applications. The real-variable theory of function spaces has always been at the core of harmonic analysis. Recently, motivated by certain questions in analysis, some more general Musielak–Orlicz Hardy-type function spaces were introduced. These spaces are defined via growth functions which may vary in both the spatial variable and the growth variable. By selecting special growth functions, the resulting spaces may have subtler and finer structures, which are necessary in order to solve various endpoint or sharp problems. This book is written for graduate students and researchers interested in function spaces and, in particular, Hardy-type spaces.
Infinite Dimensional Complex Symplectic Spaces by William Norrie Everitt,Lawrence Markus,Johannes Huebschmann Pdf
Complex symplectic spaces, defined earlier by the authors in their ""AMS Monograph"", are non-trivial generalizations of the real symplectic spaces of classical analytical dynamics. These spaces can also be viewed as non-degenerate indefinite inner product spaces, although the authors here follow the lesser known exposition within complex symplectic algebra and geometry, as is appropriate for their prior development of boundary value theory. In the case of finite dimensional complex symplectic spaces it was shown that the corresponding symplectic algebra is important for the description and classification of all self-adjoint boundary value problems for (linear) ordinary differential equations on a real interval.In later ""AMS Memoirs"" infinite dimensional complex symplectic spaces were introduced for the analysis of multi-interval systems and elliptic partial differential operators. In this current Memoir the authors present a self-contained, systematic investigation of general complex symplectic spaces, and their Lagrangian subspaces, regardless of the finite or infinite dimensionality - starting with axiomatic definitions and leading towards general Glazman-Krein-Naimark (GKN) theorems.In particular, the appropriate relevant topologies on such a symplectic space $\mathsf{S}$ are compared and contrasted, demonstrating that $\mathsf{S}$ is a locally convex linear topological space in terms of the symplectic weak topology. Also the symplectic invariants are defined (as cardinal numbers) characterizing $\mathsf{S}$, in terms of suitable Hilbert structures on $\mathsf{S}$. The penultimate section is devoted to a review of the applications of symplectic algebra to the motivating of boundary value problems for ordinary and partial differential operators. The final section, the Aftermath, is a review and summary of the relevant literature on the theory and application of complex symplectic spaces. The Memoir is completed by symbol and subject indexes.
Moduli Spaces of Polynomials in Two Variables by Javier Fernández de Bobadilla Olarzábal Pdf
In the space of polynomials in two variables $\mathbb{C}[x,y]$ with complex coefficients we let the group of automorphisms of the affine plane $\mathbb{A}^2_{\mathbb{C}}$ act by composition on the right. In this paper we investigate the geometry of the orbit space. We associate a graph with each polynomial in two variables that encodes part of its geometric properties at infinity; we define a partition of $\mathbb{C}[x,y]$ imposing that the polynomials in the same stratum are the polynomials with a fixed associated graph. The graphs associated with polynomials belong to certain class of graphs (called behaviour graphs), that has a purely combinatorial definition.We show that any behaviour graph is actually a graph associated with a polynomial. Using this we manage to give a quite precise geometric description of the subsets of the partition. We associate a moduli functor with each behaviour graph of the class, which assigns to each scheme $T$ the set of families of polynomials with the given graph parametrized over $T$. Later, using the language of groupoids, we prove that there exists a geometric quotient of the subsets of the partition associated with the given graph by the equivalence relation induced by the action of Aut$(\mathbb{C}^2)$. This geometric quotient is a coarse moduli space for the moduli functor associated with the graph. We also give a geometric description of it based on the combinatorics of the associated graph. The results presented in this memoir need the development of a certain combinatorial formalism. Using it we are also able to reprove certain known theorems in the subject.
Kahler Spaces, Nilpotent Orbits, and Singular Reduction by Johannes Huebschmann Pdf
For a stratified symplectic space, a suitable concept of stratified Kahler polarization encapsulates Kahler polarizations on the strata and the behaviour of the polarizations across the strata and leads to the notion of stratified Kahler space which establishes an intimate relationship between nilpotent orbits, singular reduction, invariant theory, reductive dual pairs, Jordan triple systems, symmetric domains, and pre-homogeneous spaces: the closure of a holomorphic nilpotent orbit or, equivalently, the closure of the stratum of the associated pre-homogeneous space of parabolic type carries a (positive) normal Kahler structure. In the world of singular Poisson geometry, the closures of principal holomorphic nilpotent orbits, positive definite hermitian JTS', and certain pre-homogeneous spaces appear as different incarnations of the same structure.The closure of the principal holomorphic nilpotent orbit arises from a semisimple holomorphic orbit by contraction. Symplectic reduction carries a positive Kahler manifold to a positive normal Kahler space in such a way that the sheaf of germs of polarized functions coincides with the ordinary sheaf of germs of holomorphic functions. Symplectic reduction establishes a close relationship between singular reduced spaces and nilpotent orbits of the dual groups.Projectivization of holomorphic nilpotent orbits yields exotic (positive) stratified Kahler structures on complex projective spaces and on certain complex projective varieties including complex projective quadrics. The space of (in general twisted) representations of the fundamental group of a closed surface in a compact Lie group or, equivalently, a moduli space of central Yang-Mills connections on a principal bundle over a surface, inherits a (positive) normal (stratified) Kahler structure. Physical examples are provided by certain reduced spaces arising from angular momentum zero.
Gromov-Hausdorff Distance for Quantum Metric Spaces/Matrix Algebras Converge to the Sphere for Quantum Gromov-Hausdorff Distance by Marc Aristide Rieffel Pdf
By a quantum metric space we mean a $C DEGREES*$-algebra (or more generally an order-unit space) equipped with a generalization of the usual Lipschitz seminorm on functions which one associates to an ordinary metric. We develop for compact quantum metric spaces a version of Gromov-Hausdorff di
Fundamental Papers in Wavelet Theory by Christopher Heil,David F. Walnut Pdf
This book traces the prehistory and initial development of wavelet theory, a discipline that has had a profound impact on mathematics, physics, and engineering. Interchanges between these fields during the last fifteen years have led to a number of advances in applications such as image compression, turbulence, machine vision, radar, and earthquake prediction. This book contains the seminal papers that presented the ideas from which wavelet theory evolved, as well as those major papers that developed the theory into its current form. These papers originated in a variety of journals from different disciplines, making it difficult for the researcher to obtain a complete view of wavelet theory and its origins. Additionally, some of the most significant papers have heretofore been available only in French or German. Heil and Walnut bring together these documents in a book that allows researchers a complete view of wavelet theory's origins and development.
Quasianalytic Monogenic Solutions of a Cohomological Equation by S. Marmi,Stefano Marmi,D. Sauzin Pdf
We prove that the solutions of a cohomological equation of complex dimension one and in the analytic category have a monogenic dependence on the parameter, and we investigate the question of their quasi analyticity. This cohomological equation is the standard linearized conjugacy equation for germs of holomorphic maps in a neighborhood of a fixed point. The parameter is the eigenvalue of the linear part, denoted by $q$. Borel's theory of non-analytic monogenic functions has been first investigated by Arnold and Herman in the related context of the problem of linearization of analytic diffeomorphisms of the circle close to a rotation.Herman raised the question whether the solutions of the cohomological equation had a quasi analytic dependence on the parameter $q$. Indeed they are analytic for $q\in\mathbb{C}\setminus\mathbb{S}^1$, the unit circle $\S^1$ appears as a natural boundary (because of resonances, i.e. roots of unity), but the solutions are still defined at points of $\mathbb{S}^1$ which lie 'far enough from resonances'. We adapt to our case Herman's construction of an increasing sequence of compacts which avoid resonances and prove that the solutions of our equation belong to the associated space of monogenic functions; some general properties of these monogenic functions and particular properties of the solutions are then studied.For instance the solutions are defined and admit asymptotic expansions at the points of $\mathbb{S}^1$ which satisfy some arithmetical condition, and the classical Carleman Theorem allows us to answer negatively to the question of quasi analyticity at these points. But resonances (roots of unity) also lead to asymptotic expansions, for which quasi analyticity is obtained as a particular case of Ecalle's theory of resurgent functions.And at constant-type points, where no quasi analytic Carleman class contains the solutions, one can still recover the solutions from their asymptotic expansions and obtain a special kind of quasi analyticity. Our results are obtained by reducing the problem, by means of Hadamard's product, to the study of a fundamental solution (which turns out to be the so-called $q$-logarithm or 'quantum logarithm'). We deduce as a corollary of our work the proof of a conjecture of Gammel on the monogenic and quasi analytic properties of a certain number-theoretical Borel-Wolff-Denjoy series.