Distributions

Distributions

by Jacques Simon
Released September 2022
Publisher(s): Wiley-ISTE
ISBN: 9781786305251

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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 sub-sequences, 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

  1. Cover
  2. Dedication
  3. Title Page
  4. Copyright
  5. Introduction
  6. Notations
  7. Chapter 1: Semi-Normed Spaces and Function Spaces
    1. 1.1. Semi-normed spaces
    2. 1.2. Comparison of semi-normed spaces
    3. 1.3. Continuous mappings
    4. 1.4. Differentiable functions
    5. 1.5. Spaces and
    6. 1.6. Integral of a uniformly continuous function
  8. Chapter 2: Space of Test Functions
    1. 2.1. Functions with compact support
    2. 2.2. Compactness in their whole of support of functions
    3. 2.3. The space
    4. 2.4. Sequential completeness of (Ω)
    5. 2.5. Comparison of (Ω) to various spaces
    6. 2.6. Convergent sequences in (Ω)
    7. 2.7. Covering by crown-shaped sets and partitions of unity
    8. 2.8. Control of the (Ω)-norms by the semi-norms of (Ω)
    9. 2.9. Semi-norms that are continuous on all the (Ω)
  9. Chapter 3: Space of Distributions
    1. 3.1. The space ′(Ω; E)
    2. 3.2. Characterization of distributions
    3. 3.3. Inclusion of (Ω; E) into ′(Ω; E)
    4. 3.4. The case where E is not a Neumann space
    5. 3.5. Measures
    6. 3.6. Continuous functions and measures
  10. Chapter 4: Extraction of Convergent Subsequences
    1. 4.1. Bounded subsets of ′(Ω; E)
    2. 4.2. Convergence in ′(Ω; E)
    3. 4.3. Sequential completeness of ′(Ω; E)
    4. 4.4. Sequential compactness in ′(Ω; E)
    5. 4.5. Change of the space E of values
    6. 4.6. The space E-weak
    7. 4.7. The space ′(Ω; E-weak) and extractability
  11. Chapter 5: Operations on Distributions
    1. 5.1. Distributions fields
    2. 5.2. Derivatives of a distribution
    3. 5.3. Image under a linear mapping
    4. 5.4. Product with a regular function
    5. 5.5. Change of variables
    6. 5.6. Some particular changes of variables
    7. 5.7. Positive distributions
    8. 5.8. Distributions with values in a product space
  12. Chapter 6: Restriction, Gluing and Support
    1. 6.1. Restriction
    2. 6.2. Additivity with respect to the domain
    3. 6.3. Local character
    4. 6.4. Localization-extension
    5. 6.5. Gluing
    6. 6.6. Annihilation domain and support
    7. 6.7. Properties of the annihilation domain and support
    8. 6.8. The space
  13. Chapter 7: Weighting
    1. 7.1. Weighting by a regular function
    2. 7.2. Regularizing character of the weighting by a regular function
    3. 7.3. Derivatives and support of distributions weighted by a regular weight
    4. 7.4. Continuity of the weighting by a regular function
    5. 7.5. Weighting by a distribution
    6. 7.6. Comparison of the definitions of weighting
    7. 7.7. Continuity of the weighting by a distribution
    8. 7.8. Derivatives and support of a weighted distribution
    9. 7.9. Miscellanous properties of weighting
  14. Chapter 8: Regularization and Applications
    1. 8.1. Local regularization
    2. 8.2. Properties of local approximations
    3. 8.3. Global regularization
    4. 8.4. Convergence of global approximations
    5. 8.5. Properties of global approximations
    6. 8.6. Commutativity and associativity of weighting
    7. 8.7. Uniform convergence of sequences of distributions
  15. Chapter 9: Potentials and Singular Functions
    1. 9.1. Surface integral over a sphere
    2. 9.2. Distribution associated with a singular function
    3. 9.3. Derivatives of a distribution associated with a singular function
    4. 9.4. Elementary Newtonian potential
    5. 9.5. Newtonian potential of order n
    6. 9.6. Localized potential
    7. 9.7. Dirac mass as derivatives of continuous functions
    8. 9.8. Heaviside potential
    9. 9.9. Weighting by a singular weight
  16. Chapter 10: Line Integral of a Continuous Field
    1. 10.1. Line integral along a 1 path
    2. 10.2. Change of variable in a path
    3. 10.3. Line integral along a piecewise 1 path
    4. 10.4. The homotopy invariance theorem
    5. 10.5. Connectedness and simply connectedness
  17. Chapter 11: Primitives of Functions
    1. 11.1. Primitive of a function field with a zero line integral
    2. 11.2. Tubular flows and concentration theorem
    3. 11.3. The orthogonality theorem for functions
    4. 11.4. Poincaré’s theorem
  18. Chapter 12: Properties of Primitives of Distributions
    1. 12.1. Representation by derivatives
    2. 12.2. Distribution whose derivatives are zero or continuous
    3. 12.3. Uniqueness of a primitive
    4. 12.4. Locally explicit primitive
    5. 12.5. Continuous primitive mapping
    6. 12.6. Harmonic distributions, distributions with a continuous Laplacian
  19. Chapter 13: Existence of Primitives
    1. 13.1. Peripheral gluing
    2. 13.2. Reduction to the function case
    3. 13.3. The orthogonality theorem
    4. 13.4. Poincaré’s generalized theorem
    5. 13.5. Current of an incompressible two dimensional field
    6. 13.6. Global versus local primitives
    7. 13.7. Comparison of the existence conditions of a primitive
    8. 13.8. Limits of gradients
  20. Chapter 14: Distributions of Distributions
    1. 14.1. Characterization
    2. 14.2. Bounded sets
    3. 14.3. Convergent sequences
    4. 14.4. Extraction of convergent subsequences
    5. 14.5. Change of the space of values
    6. 14.6. Distributions of distributions with values in E-weak
  21. Chapter 15: Separation of Variables
    1. 15.1. Tensor products of test functions
    2. 15.2. Decomposition of test functions on a product of sets
    3. 15.3. The tensorial control theorem
    4. 15.4. Separation of variables
    5. 15.5. The kernel theorem
    6. 15.6. Regrouping of variables
    7. 15.7. Permutation of variables
  22. Chapter 16: Banach Space Valued Distributions
    1. 16.1. Finite order distributions
    2. 16.2. Weighting of a finite order distribution
    3. 16.3. Finite order distribution as derivatives of continuous functions
    4. 16.4. Finite order distribution as derivative of a single function
    5. 16.5. Distributions in a Banach space as derivatives of functions
    6. 16.6. Non-representability of distributions with values in a Fréchet space
    7. 16.7. Extendability of distributions with values in a Banach space
    8. 16.8. Cancellation of distributions with values in a Banach space
  23. Appendix: Reminders
    1. A.1. Notation and numbering
    2. A.2. Semi-normed spaces
    3. A.3. Continuous mappings, duality
    4. A.4. Continuous or differentiable functions
    5. A.5. Integration of uniformly continuous functions
  24. Bibliography
  25. Index
  26. Wiley End User License Agreement

Product information

  • Title: Distributions
  • Author(s): Jacques Simon
  • Release date: September 2022
  • Publisher(s): Wiley-ISTE
  • ISBN: 9781786305251