Antosik P. Theory of Distributions

Amsterdam: Elsevier, 1973. — 285 p.The book uses a sequential approach to introduce the theory of distributions, focusing on the properties of sequences of functions and their limits, provides a definition for the product of distributions using a delta sequence.Preface
Elementary theory of distributions of a single real variable
Fundamental definitions
The identification principle
Fundamental sequences of continuous functions
The definition of distributions
Distributions as a generalization of the notion of functions
Operations on distributions
Algebraic operations on distributions
Derivation of distributions
The definition of distributions by derivatives
Locally integrable functions
Sequences and series of distributions
Distributions depending on a continuous parameter
Multiplication of distributions by functions
Compositions
Local properties
Equality of distributions in intervals
Functions with poles
Derivative as the limit of a difference quotient
The value of a distribution at a point
Existence theorems for values of distributions
The value of a distribution at infinity
Extension of the theory
The integral of a distribution
Periodic distributions
Distributions of infinite order
Elementary theory of distributions of several real variables
Fundamental definitions
Terminology and notation
Uniform and almost uniform convergence
Fundamental sequences of smooth functions
The definition of distributions
Operations on distributions
Multiplication by a number
Addition
Regular operations
Subtraction, translation, derivation
Multiplication of a distribution by a smooth function
Substitution
Product of distributions with separated variables
Convolution with a smooth function vanishing outside an interval
Calculations with distributions
Local properties
Delta-sequences and the delta-distribution
Distributions in subsets
Distributions as a generalization of the notion of continuous functions...
Operations on continuous functions
Locally integrable functions
Operations on locally integrable functions
Sequences of distributions
Convergence and regular operations
Distributional^ convergent sequences of smooth functions
Locally convergent sequences of distributions
Extension of the theory
Distributions depending on a continuous parameter
Multidimensional substitution
Distributions constant in some variables
Dimension of distributions
Distributions with vanishing mth derivatives
Advanced theory of distributions
Convolution
Convolution of two functions
Convolution of three functions
Associativity of convolution
Convolution of a locally integrable function with a smooth function of bounded carrier
Delta-sequences and regular sequences
Delta-sequences
Regular sequences
Convolution of a convergent sequence with a delta-sequence
Existence theorems for convolutions
Convolutive "dual sets
Convolution of functions with compatible carriers
Properties of compatible sets
Associativity of convolution of functions with restricted carriers
A particular case
Convolution of two smooth functions
Square integrable functions
Fundamental definitions and theorems
Regular sequences
The Fourier transform of square integrable functions
Two approximation theorems
The main approximation theorem
Hermite polynomials of a real variable
Hermite polynomials of several variables
Series of Hermite functions
The Fourier transform of an Hermite expansion
Inner product
Inner product of two functions
Inner product of three functions
Convolution of distributions
Distributions of finite order
Convolution of a distribution with a smooth function of bounded carrier
Convolution of two distributions
Convolution of distributions with compatible carriers
Tempered distributions
Tempered derivatives.
Tempered integrals
Tempered distributions
Subclasses of tempered distributions
Tempered convergence of sequences
Inner product with a smooth function of bounded carrier
Fundamental sequences and distributions in R°
Proof of the regularity of inner product
The space of rapidly decreasing smooth functions
Extension of the definition of an inner product
Tempered Hermite series
Hermite series and their derivatives
Square integrable functions and rapidly decreasing functions
Examples and remarks
Multidimensional expansions
Some particular expansions
The Fourier transform
An analogy with power series
The Fourier transform of a convolution
Periodic distributions
Smooth integral
Integral over the period
Decomposition theorem for periodic distributions
Periodic inner product
Periodic convolution
Expansions in Fourier series
The Fourier transform of periodic distributions
The Kothe spaces
General remarks
Spaces of sequences
Kothe's echelon space and co-echelon space
Strong and weak boundedness
Diagonal Theorem
The proof of the Boundedness Theorem
Strong convergence and weak convergence
A more general formulation of the theory
Functionals on the space of rapidly decreasing matrices
Applications of the Kothe spaces
Applications to tempered distributions
Convergence in ^ and M
Tempered distributions as functionals
Application to arbitrary distributions...
Distributions as functionals
Application to periodic distributions
Periodic distributions as functionals
Applications of the equivalence of weak and strong convergence
Convergence and regular operations
The value of a distribution at a point
Properties of the delta-distribution
Product of two' distributions
Non existence of δ^2
The product x.1/x
On the associativity of the product
The Hilbert transform and its applications
The Hilbert transform
Non existence of (1/x)^2
Some formulae for the Hilbert transform.
The product 1/x.δ
On the equation xf = δ
Generalization to several variables
Applications of the Fourier transform
The convolution 1/x • 1/x
The square of δ+1/π^2 • 1/x
The formula δ^2 - 1/π^2 • (1/x)^2 = - 1/π^2 • 1/x^2
Final remarks
Generalized operations
A system of differential equations
Some remarks on integrals of distributions
Distributions with a one-point carrier
Appendix
Induction
Recursive definition
Examples
Finite induction
Newton's symbol in the multidimensional case
The formulae of Leibniz and of Schwartz
Bibliography
Index of Authors and Terminology