THEORY OF DISTRIBUTIONS

Transcription

1 THEORY OF DISTRIBUTIONS THE SEQUENTIAL APPROACH by PIOTR ANTOSIK Special Research Centre of the Polish Academy of Sciences in Katowice JAN MIKUSltfSKI Special Research Centre of the Polish Academy of Sciences in Katowice ROMAN SIKORSKI University of Warsaw ELSEVIER SCIENTIFIC PUBLISHING COMPANY AMSTERDAM PWN POLISH SCIENTIFIC PUBLISHERS WARSZAWA 1973

2 Contents Preface xii Part I Elementary theory of distributions of a single real variable Introduction to Part I 3 1. 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 f 54 Part II Elementary theory of distributions of several real variables Introduction to Part II Fundamental definitions Terminology and notation 61

3 Vill CONTENTS 1.2. 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.' 104 Part III Advanced theory of distributions Introduction to Part III Convolution Ill 1.1. Convolution of two functions Ill 1.2. Convolution of three functions Associativity of convolution Convolution of a locally mtegrable 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 119

4 CONTENTS ' IX 3. 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 195

5 X CONTENTS 8.7. 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 r 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 d The product x 244 x On the associativity of the product The Hilbert transform and its applications The Hilbert transform,.... ; 246 / 1 \ Non existence of I I Some formulae for the Hilbert transform The product d 249 x

6 CONTENTS XI On the equation xf = Generalization to several variables Applications of the Fourier transform The convolution * 253 x x The square of 8+^ 253 TZ 2 X The formula S 2 ~l I = ^-V 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.... y Examples Finite induction Newton's symbol in the multidimensional case The formulae of Leibniz and of Schwartz 265 Bibliography 268 Index of Authors and Terminology 271