Smooth Molecular Decompositions of Functions and Singular Integral Operators

Smooth Molecular Decompositions of Functions and Singular Integral Operators PDF

Author: John E. Gilbert

Publisher: American Mathematical Soc.

Published: 2002

Total Pages: 89

ISBN-13: 0821827723

DOWNLOAD EBOOK →

Under minimal assumptions on a function $\psi$ the authors obtain wavelet-type frames of the form $\psi_{j, k}(x) = r DEGREES{(1/2)n j} \psi(r DEGREESj x - sk), j \in \integer, k \in \integer DEGREESn, $ for some $r > 1$ and $s > 0$. This collection is shown to be a frame for a scale of Triebel-Lizorkin spaces (which includes Lebesgue, Sobolev and Hardy spaces) and the reproducing formula converges in norm as well as pointwise a.e. The construction follows from a characterization of those operators which are bounded on a space of smooth molecules. This characterization also allows us to decompose a broad range of singular integral operators in ter

Pointwise Variable Anisotropic Function Spaces on Rn

Pointwise Variable Anisotropic Function Spaces on Rn PDF

Author: Shai Dekel

Publisher: Walter de Gruyter GmbH & Co KG

Published: 2022-04-04

Total Pages: 250

ISBN-13: 3110761793

DOWNLOAD EBOOK →

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.

Time‒Frequency and Time‒Scale Methods

Time‒Frequency and Time‒Scale Methods PDF

Author: Jeffrey A. Hogan

Publisher: Springer Science & Business Media

Published: 2007-12-21

Total Pages: 403

ISBN-13: 0817644318

DOWNLOAD EBOOK →

Developed in this book are several deep connections between time-frequency (Fourier/Gabor) analysis and time-scale (wavelet) analysis, emphasizing the powerful adaptive methods that emerge when separate techniques from each area are properly assembled in a larger context. While researchers at the forefront of these areas are well aware of the benefits of such a unified approach, there remains a knowledge gap in the larger community of practitioners about the precise strengths and limitations of Fourier/Gabor analysis versus wavelets. This book fills that gap by presenting the interface of time-frequency and time-scale methods as a rich area of work. "Foundations of Time-Frequency and Time-Scale Methods" will be suitable for applied mathematicians and engineers in signal/image processing and communication theory, as well as researchers and students in mathematical analysis, signal analysis, and mathematical physics.

Twentieth Century Harmonic Analysis

Twentieth Century Harmonic Analysis PDF

Author: J.S. Byrnes

Publisher: Springer Science & Business Media

Published: 2001-09-30

Total Pages: 428

ISBN-13: 9780792371687

DOWNLOAD EBOOK →

Almost a century ago, harmonic analysis entered a (still continuing) Golden Age, with the emergence of many great masters throughout Europe. They created a wealth of profound analytic methods, to be successfully exploited and further developed by succeeding generations. This flourishing of harmonic analysis is today as lively as ever, as the papers presented here demonstrate. In addition to its own ongoing internal development and its basic role in other areas of mathematics, physics and chemistry, financial analysis, medicine, and biological signal processing, harmonic analysis has made fundamental contributions to essentially all twentieth century technology-based human endeavours, including telephone, radio, television, radar, sonar, satellite communications, medical imaging, the Internet, and multimedia. This ubiquitous nature of the subject is amply illustrated. The book not only promotes the infusion of new mathematical tools into applied harmonic analysis, but also to fuel the development of applied mathematics by providing opportunities for young engineers, mathematicians and other scientists to learn more about problem areas in today's technology that might benefit from new mathematical insights.

Boundedness Results for Operators with Singular Kernels on Distribution Spaces

Boundedness Results for Operators with Singular Kernels on Distribution Spaces PDF

Author: Rodolfo Humberto Torres

Publisher: American Mathematical Soc.

Published: 1991

Total Pages: 185

ISBN-13: 0821825054

DOWNLOAD EBOOK →

In this monograph, the author generalizes the T1 theorem of G. David and J.-L. Journé to the context of Triebel-Lizorkin spaces, which include most of the function and distribution spaces of importance in classical analysis. M. Frazier and B. Jawerth have shown that these spaces admit decompositions in terms of certain fundamental building blocks, known as "smooth atoms'' and "smooth molecules''. In the author's words, "to prove that an operator is bounded on a Triebel-Lizorkin space, it is enough to show that it maps every family of smooth atoms into a family of smooth molecules''. The same basic techniques can be used to study operators between two different Triebel-Lizorkin spaces. Results are obtained for a wide variety of operators acting on the Triebel-Lizorkin spaces, including generalized Calderón-Zygmund operators and their derivatives; potential operators; fractional integral operators; and the Hörmander classes of pseudodifferential operators. When these general results are restricted to specific spaces, many classical boundedness results are recovered.

Approximation and Entropy Numbers of Volterra Operators with Application to Brownian Motion

Approximation and Entropy Numbers of Volterra Operators with Application to Brownian Motion PDF

Author: Mikhail Anatolʹevich Lifshit︠s︡

Publisher: American Mathematical Soc.

Published: 2002

Total Pages: 103

ISBN-13: 082182791X

DOWNLOAD EBOOK →

This text considers a specific Volterra integral operator and investigates its degree of compactness in terms of properties of certain kernel functions. In particular, under certain optimal integrability conditions the entropy numbers $e_n(T_{\rho, \psi})$ satisfy $c_1\norm{\rho\psi}_r0$.

Elliptic Partial Differential Operators and Symplectic Algebra

Elliptic Partial Differential Operators and Symplectic Algebra PDF

Author: William Norrie Everitt

Publisher: American Mathematical Soc.

Published: 2003

Total Pages: 130

ISBN-13: 0821832352

DOWNLOAD EBOOK →

This investigation introduces a new description and classification for the set of all self-adjoint operators (not just those defined by differential boundary conditions) which are generated by a linear elliptic partial differential expression $A(\mathbf{x}, D)=\sum_{0\, \leq\, \left s\right \, \leq\,2m}a_{s} (\mathbf{x})D DEGREES{s}\;\text{for all}\;\mathbf{x}\in\Omega$ in a region $\Omega$, with compact closure $\overline{\Omega}$ and $C DEGREES{\infty }$-smooth boundary $\partial\Omega$, in Euclidean space $\mathbb{E} DEGREES{r}$ $(r\geq2).$ The order $2m\geq2$ and the spatial dimensio

Lanterna Rally - Theme by Grace Themes