Book description
This book presents a simple and original theory of distributions, both real and vector, adapted to the study of partial differential equations. It deals with value distributions in a Neumann space, that is, in which any Cauchy suite converges, which encompasses the Banach and Fréchet spaces and the same “weak” spaces. Alongside the usual operations – derivation, product, variable change, variable separation, restriction, extension and regularization – Distributions presents a new operation: weighting.This operation produces properties similar to those of convolution for distributions defined in any open space. Emphasis is placed on the extraction of convergent subsequences, the existence and study of primitives and the representation by gradient or by derivatives of continuous functions. Constructive methods are used to make these tools accessible to students and engineers.
Table of contents
 Cover
 Dedication
 Title Page
 Copyright
 Introduction
 Notations
 Chapter 1: SemiNormed Spaces and Function Spaces

Chapter 2: Space of Test Functions
 2.1. Functions with compact support
 2.2. Compactness in their whole of support of functions
 2.3. The space
 2.4. Sequential completeness of (Ω)
 2.5. Comparison of (Ω) to various spaces
 2.6. Convergent sequences in (Ω)
 2.7. Covering by crownshaped sets and partitions of unity
 2.8. Control of the (Ω)norms by the seminorms of (Ω)
 2.9. Seminorms that are continuous on all the (Ω)
 Chapter 3: Space of Distributions
 Chapter 4: Extraction of Convergent Subsequences
 Chapter 5: Operations on Distributions
 Chapter 6: Restriction, Gluing and Support

Chapter 7: Weighting
 7.1. Weighting by a regular function
 7.2. Regularizing character of the weighting by a regular function
 7.3. Derivatives and support of distributions weighted by a regular weight
 7.4. Continuity of the weighting by a regular function
 7.5. Weighting by a distribution
 7.6. Comparison of the definitions of weighting
 7.7. Continuity of the weighting by a distribution
 7.8. Derivatives and support of a weighted distribution
 7.9. Miscellanous properties of weighting
 Chapter 8: Regularization and Applications

Chapter 9: Potentials and Singular Functions
 9.1. Surface integral over a sphere
 9.2. Distribution associated with a singular function
 9.3. Derivatives of a distribution associated with a singular function
 9.4. Elementary Newtonian potential
 9.5. Newtonian potential of order n
 9.6. Localized potential
 9.7. Dirac mass as derivatives of continuous functions
 9.8. Heaviside potential
 9.9. Weighting by a singular weight
 Chapter 10: Line Integral of a Continuous Field
 Chapter 11: Primitives of Functions
 Chapter 12: Properties of Primitives of Distributions

Chapter 13: Existence of Primitives
 13.1. Peripheral gluing
 13.2. Reduction to the function case
 13.3. The orthogonality theorem
 13.4. Poincaré’s generalized theorem
 13.5. Current of an incompressible two dimensional field
 13.6. Global versus local primitives
 13.7. Comparison of the existence conditions of a primitive
 13.8. Limits of gradients
 Chapter 14: Distributions of Distributions
 Chapter 15: Separation of Variables

Chapter 16: Banach Space Valued Distributions
 16.1. Finite order distributions
 16.2. Weighting of a finite order distribution
 16.3. Finite order distribution as derivatives of continuous functions
 16.4. Finite order distribution as derivative of a single function
 16.5. Distributions in a Banach space as derivatives of functions
 16.6. Nonrepresentability of distributions with values in a Fréchet space
 16.7. Extendability of distributions with values in a Banach space
 16.8. Cancellation of distributions with values in a Banach space
 Appendix: Reminders
 Bibliography
 Index
 Wiley End User License Agreement
Product information
 Title: Distributions
 Author(s):
 Release date: September 2022
 Publisher(s): WileyISTE
 ISBN: 9781786305251
You might also like
book
Mathematical Foundations of Fuzzy Sets
Mathematical Foundations of Fuzzy Sets Introduce yourself to the foundations of fuzzy logic with this easytouse …
book
The Kitchen Pantry Scientist Math for Kids
The Kitchen Pantry Scientist: Math for Kids features biographies of 20+ leading mathematicians, past and present, …
book
In the Line of Fire, 3rd Edition
In this fullyupdated edition of his classic In the Line of Fire, the world’s #1 presentation …
book
Noise Control, 2nd Edition
The second edition of Noise Control: From Concept to Application, newly expanded and thoroughly updated, now …