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
Contents Preface xii Part I Elementary theory of distributions of a single real variable Introduction to Part I 3 1. Fundamental definitions 5 1.1. The identification principle 5 1.2. Fundamental sequences of continuous functions 6 1.3. The definition of distributions 10 1.4. Distributions as a generalization of the notion of functions 12 2. Operations on distributions 14 2.1. Algebraic operations on distributions 14 2.2. Derivation of distributions 15 2.3. The definition of distributions by derivatives 18 2.4. Locally integrable functions 20 2.5. Sequences and series of distributions 22 2.6. Distributions depending on a continuous parameter 25 2.7. Multiplication of distributions by functions 27 2.8. Compositions 30 3. Local properties 33 3.1. Equality of distributions in intervals 33 3.2. Functions with poles 35 3.3. Derivative as the limit of a difference quotient 36 3.4. The value of a distribution at a point 38 3.5. Existence theorems for values of distributions 40 3.6. The value of a distribution at infinity 44 4. Extension of the theory 46 4.1. The integral of a distribution 46 4.2. Periodic distributions. 49 4.3. Distributions of infinite order f 54 Part II Elementary theory of distributions of several real variables Introduction to Part II 59 1. Fundamental definitions 61 1.1. Terminology and notation 61
Vill CONTENTS 1.2. Uniform and almost uniform convergence 63 1.3. Fundamental sequences of smooth functions 63 1.4. The definition of distributions 64 2. Operations on distributions 66 2.1. Multiplication by a number 66 2.2. Addition 66 2.3. Regular operations 67 2.4. Subtraction, translation, derivation 68 2.5. Multiplication of a distribution by a smooth function 69 2.6. Substitution 70 2.7. Product of distributions with separated variables 71 2.8. Convolution with a smooth function vanishing outside an interval 72 2.9. Calculations with distributions 73 3. Local properties 75 3.1. Delta-sequences and the delta-distribution 75 3.2. Distributions in subsets 77 3.3. Distributions as a generalization of the notion of continuous functions... 78 3.4. Operations on continuous functions 80 3.5. Locally integrable functions 82 3.6. Operations on locally integrable functions 84 3.7. Sequences of distributions 86 3.8. Convergence and regular operations 89 3.9. Distributional^ convergent sequences of smooth functions 91 3.10. Locally convergent sequences of distributions 93 4. Extension of the theory 96 4.1. Distributions depending on a continuous parameter 96 4.2. Multidimensional substitution 97 4.3. Distributions constant in some variables 99 4.4. Dimension of distributions 101 4.5. Distributions with vanishing mth derivatives.' 104 Part III Advanced theory of distributions Introduction to Part III 109 1. Convolution Ill 1.1. Convolution of two functions Ill 1.2. Convolution of three functions 112 1.3. Associativity of convolution 113 1.4. Convolution of a locally mtegrable function with a smooth function of bounded carrier 114 2. Delta-sequences and regular sequences 116 2.1. Delta-sequences 116 2.2. Regular sequences 117 2.3. Convolution of a convergent sequence with a delta-sequence 119
CONTENTS ' IX 3. Existence theorems for convolutions 121 3.1. Convolutive "dual sets 121 3.2. Convolution of functions with compatible carriers 124 3.3. Properties of compatible sets 125 3.4. Associativity of convolution of functions with restricted carriers 127 3.5. A particular case 129 3.6. Convolution of two smooth functions 130 4. Square integrable functions 132 4.1. Fundamental definitions and theorems 132 4.2. Regular sequences 133 4.3. The Fourier transform of square integrable functions 135 4.4. Two approximation theorems 137 4.5. The main approximation theorem 140 4.6. Hermite polynomials of a real variable 141 4.7. Hermite polynomials of several variables 142 4.8. Series of Hermite functions 143 4.9. The Fourier transform of an Hermite expansion 146 5. Inner product 148 5.1. Inner product of two functions 148 5.2. Inner product of three functions 149 6. Convolution of distributions 151 6.1. Distributions of finite order 151 6.2. Convolution of a distribution with a smooth function of bounded carrier. 152 6.3. Convolution of two distributions 153 6.4. Convolution of distributions with compatible carriers 156 7. Tempered distributions - 161 7.1. Tempered derivatives. 161 7.2. Tempered integrals 162 7.3. Tempered distributions 165 7.4. Subclasses of tempered distributions 166 7.5. Tempered convergence of sequences 169 7.6. Inner product with a smooth function of bounded carrier........ 173 7.7. Fundamental sequences and distributions in R 173 7.8. Proof of the regularity of inner product 174 7.9. The space of rapidly decreasing smooth functions 175 7.10. Extension of the definition of an inner product 178 8. Tempered Hermite series / 180 8.1. Hermite series and their derivatives 180 8.2. Square integrable functions and rapidly decreasing functions 183 8.3. Examples and remarks 189 8.4. Multidimensional expansions 191 8.5. Some particular expansions 193 8.6. The Fourier transform 195
X CONTENTS 8.7. An analogy with power series 198 8.8. The Fourier transform of a convolution 199 9. Periodic distributions 201 9.1. Smooth integral 201 9.2. Integral over the period 203 9.3. Decomposition theorem for periodic distributions 204 9.4. Periodic inner product 206 9.5. Periodic convolution 208 9.6. Expansions in Fourier series 210 9.7. The Fourier transform of periodic distributions 214 10. The Kothe spaces 215 10.1. General remarks 215 10.2. Spaces of sequences 215 10.3. Kothe's echelon space and co-echelon space 216 10.4. Strong and weak boundedness 217 10.5. Diagonal Theorem 217 10.6. The proof of the Boundedness Theorem 220 10.7. Strong convergence and weak convergence 221 10.8. A more general formulation of the theory 222 10.9. Functionals on the space of rapidly decreasing matrices 224 11. Applications of the Kothe spaces 226 11.1 Applications to tempered distributions 226 11.2. Convergence in ^ and M 228 11.3. Tempered distributions as functionals 230 11.4. Application to arbitrary distributions... 232 11.5. Distributions as functionals 233 11.6. Application to periodic distributions 235 11.7. Periodic distributions as functionals r 238 12. Applications of the equivalence of weak and strong convergence 239 12.1. Convergence and regular operations 239 12.2. The value of a distribution at a point 240 12.3. Properties of the delta-distribution 242 12.4. Product of two' distributions 242 12.5. Non existence of d 2 243 12.6. The product x 244 x 12.7. On the associativity of the product 245 13. The Hilbert transform and its applications 246 13.1. The Hilbert transform,.... ; 246 / 1 \ 2 13.2. Non existence of I I 247 13.3. Some formulae for the Hilbert transform. 248 13.4. The product d 249 x
CONTENTS XI 13.5. On the equation xf = 6. 251 13.6. Generalization to several variables 252 14. Applications of the Fourier transform 253 14.1. The convolution * 253 x x 14.2. The square of 8+^ 253 TZ 2 X 14.3. The formula S 2 ~l I = ^-V 254 15. Final remarks 256 15.1. Generalized operations 256 15.2. A system of differential equations 257 15.3. Some remarks on integrals of distributions 258 15.4. Distributions with a one-point carrier 259 16. Appendix 261 16.1. Induction 261 16.2. Recursive definition.... y 262 16.3. Examples 263 16.4. Finite induction 264 16.5. Newton's symbol in the multidimensional case 264 16.6. The formulae of Leibniz and of Schwartz 265 Bibliography 268 Index of Authors and Terminology 271