[chapter] [chapter] [chapter] 1 2 Introduction to Hyperfunctions and Their Integral Transforms Urs E. Graf 2009 ii Contents Preface ix 1 Introduction to Hyperfunctions 1 1.1 Generalized Functions . 1 1.2 The Concept of a Hyperfunction . 3 1.3 Properties of Hyperfunctions . 13 1.3.1 Linear Substitution . 13 1.3.2 Hyperfunctions of the Type f(φ(x)) . 15 1.3.3 Differentiation . 18 1.3.4 The Shift Operator as a Differential Operator . 25 1.3.5 Parity, Complex Conjugate and Realness . 25 1.3.6 The Equation φ(x)f(x) = h(x)............... 28 1.4 Finite Part Hyperfunctions . 33 1.5 Integrals . 37 1.5.1 Integrals with respect to the Independent Variable . 37 1.5.2 Integrals with respect to a Parameter . 43 1.6 More Familiar Hyperfunctions . 44 1.6.1 Unit-Step, Delta Impulses, Sign, Characteristic Hyper- functions . 44 1.6.2 Integral Powers . 45 1.6.3 Non-integral Powers . 49 1.6.4 Logarithms . 51 1.6.5 Upper and Lower Hyperfunctions . 55 α 1.6.6 The Normalized Power x+=Γ(α + 1) . 58 1.6.7 Hyperfunctions Concentrated at One Point . 61 2 Analytic Properties 63 2.1 Sequences, Series, Limits . 63 2.2 Cauchy-type Integrals . 71 2.3 Projections of Functions . 75 2.3.1 Functions satisfying the H¨olderCondition . 77 2.3.2 Projection Theorems . 78 2.3.3 Convergence Factors . 87 2.3.4 Homologous and Standard Hyperfunctions . 88 2.4 Projections of Hyperfunctions . 91 2.4.1 Holomorphic and Meromorphic Hyperfunctions . 91 2.4.2 Standard Defining Functions . 94 2.4.3 Micro-analytic Hyperfunctions . 110 iii 2.4.4 Support, Singular Support and Singular Spectrum . 110 2.5 Product of Hyperfunctions . 113 2.5.1 Product of Upper or Lower Hyperfunctions . 114 2.5.2 Products in the Case of Disjoint Singular Supports . 115 2.5.3 The Integral of a Product . 119 2.5.4 Hadamard's Finite Part of an Integral . 125 2.6 Periodic Hyperfunctions and Their Fourier Series . 128 2.7 Convolutions of Hyperfunctions . 137 2.7.1 Definition and Existence of the Convolution . 137 2.7.2 Sufficient Conditions for the Existence of Convolutions . 141 2.7.3 Operational Properties . 144 2.7.4 Principal Value Convolution . 149 2.8 Integral Equations I . 151 3 Laplace Transforms 155 3.1 Loop Integrals . 155 3.2 The Two-Sided Laplace Transform . 159 3.2.1 The Classical Laplace Transform . 159 3.3 Laplace Transforms of Hyperfunctions . 162 3.4 Transforms of some Familiar Hyperfunctions . 171 3.4.1 Dirac Impulses and their Derivatives . 171 3.4.2 Non negative Integral Powers . 173 3.4.3 Negative Integral Powers . 174 3.4.4 Non integral Powers . 174 3.4.5 Powers with Logarithms . 175 3.4.6 Exponential Integrals . 177 3.4.7 Transforms of Finite Part Hyperfunctions . 182 3.5 Operational Properties . 188 3.5.1 Linearity . 188 3.5.2 Image Translation Rule . 189 3.5.3 The Multiplication or Image Differentiation Rule . 192 3.5.4 Similarity Rule . 193 3.5.5 Differentiation Rule . 194 3.5.6 Integration Rule . 198 3.5.7 Original Translation Rule . 200 3.5.8 Linear Substitution Rules . 201 3.6 Inverse Laplace Transforms and Convolutions . 203 3.6.1 Inverse Laplace Transforms . 203 3.6.2 The Convolution Rule . 214 3.6.3 Fractional Integrals and Derivatives . 222 3.7 Right-sided Laplace Transforms . 223 3.8 Integral Equations II . 226 3.8.1 Volterra Integral Equations of Convolution Type . 226 3.8.2 Convolution Integral Equations over an Infinite Range . 231 4 Fourier Transforms 239 4.1 Fourier Transforms of Hyperfunctions . 239 4.1.1 Basic Definitions . 239 4.1.2 Connection to Laplace Transformation . 242 4.2 Fourier Transforms of Some Familiar Hyperfunctions . 244 iv 4.3 Inverse Fourier Transforms . 249 4.3.1 Reciprocity . 252 4.4 Operational Properties . 253 4.4.1 Linear Substitution Rule . 254 4.4.2 Shift-Rules . 255 4.4.3 Complex Conjugation and Realness . 255 4.4.4 Differentiation and Multiplication Rule . 256 4.4.5 Convolution Rules . 258 4.5 Further Examples . 263 4.6 Poisson's Summation Formula . 264 4.7 Application to Integral and Differential Equations . 267 4.7.1 Integral Equations III . 268 4.7.2 Heat Equation and Weierstrass Transformation . 270 5 Hilbert Transforms 273 5.1 Hilbert Transforms of Hyperfunctions . 273 5.1.1 Definition and Basic Properties . 273 5.1.2 Operational Properties . 287 5.1.3 Using Fourier Transforms . 290 5.2 Analytic Signals and Conjugate Hyperfunctions . 293 5.3 Integral Equations IV . 297 6 Mellin Transforms 307 6.1 The Classical Mellin Transformation . 307 6.2 Mellin Transforms of Hyperfunctions . 311 6.3 Operational Properties . 313 6.3.1 Linearity . 313 6.3.2 Scale Changes . 313 6.3.3 Multiplication by (log x)n .................. 314 6.3.4 Multiplication by xµ; µ 2 C ................. 315 6.3.5 Reflection . 315 6.3.6 Differentiation Rules . 316 6.3.7 Integration Rules . 317 6.4 Inverse Mellin Transformation . 318 6.5 M-Convolutions . 320 6.5.1 Reciprocal Integral Transforms . 323 6.5.2 Transform of a Product and Parseval's Formula . 324 6.6 Applications . 325 6.6.1 Dirichlet's Problem in a Wedge-shaped Domain . 325 6.6.2 Euler's Differential Equation . 327 6.6.3 Integral Equations V . 329 6.6.4 Summation of Series . 330 7 Hankel Transforms 335 7.1 Hankel Transforms of Ordinary Functions . 335 7.1.1 Genesis of the Hankel Transform . 335 7.1.2 Cylinder Functions . 338 7.1.3 Lommel's Integral . 344 7.1.4 MacRobert's Proof . 346 7.1.5 Some Hankel Transforms of Ordinary Functions . 347 v 7.1.6 Operational Properties . 350 7.2 Hankel Transforms of Hyperfunctions . 354 7.2.1 Basic Definitions . 354 7.2.2 Transforms of some Familiar Hyperfunctions . 356 7.2.3 Operational Properties . 361 7.3 Applications . 365 A Complements 371 A.1 Physical Interpretation of Hyperfunctions . 371 A.1.1 Flow Fields and Holomorphic Functions . 371 A.1.2 P´olya fields and Defining Functions . 373 A.2 Laplace Transforms in the Complex Plane . 375 A.2.1 Functions of Exponential Type . 375 A.2.2 Laplace Hyperfunctions and their Transforms . 384 A.3 Some Basic Theorems of Function Theory . 387 A.3.1 Interchanging Infinite Series with Improper Integrals . 387 A.3.2 Reversing the Order of Integration . 389 A.3.3 Defining Holomorphic Functions by Series and Integrals . 389 B Tables 393 B.0.4 Convolution Properties of Hyperfunctions . 393 B.0.5 Operational Rules for the Laplace Transformation . 393 B.0.6 Some Laplace Transforms of Hyperfunctions . 394 B.0.7 Operational Rules for the Fourier Transformation . 396 B.0.8 Some Fourier Transforms of Hyperfunctions . 396 B.0.9 Operational Rules for the Hilbert Transformation . 398 B.0.10 Some Hilbert Transforms of Hyperfunctions . 399 B.0.11 Operational Rules for the Mellin Transformation . 400 B.0.12 Some Mellin Transforms of Hyperfunctions . 401 B.0.13 Operational Rules for the Hankel Transformation . 402 B.0.14 Some Hankel Transforms of order ν of Hyperfunctions . 403 B.1 List of Symbols . 404 vi List of Figures 1.1 Complex neighborhood D(I).................... 3 1.2 Intuitive picture of the Dirac impulse . 8 1.3 Intuitive picture of the unit-step . 9 1.4 Intuitive picture of the first derivative of the Dirac impulse . 19 1.5 Intuitive picture of the second derivative of the Dirac impulse . 21 1.6 Piecewise real analytic function. 22 1.7 Intuitive picture of fp(1/x) . 35 1.8 Intuitive picture of fp(1=(x − a)2 .................. 36 + − 1.9 Contours γa;b and γa;b ........................ 38 1.10 Closed contour γ = (a; b)....................... 39 1.11 Contour for Example 1.29 . 40 1.12 Integration loop (−a; 1)....................... 42 1.13 Integration loop (−∞; as)...................... 42 1.14 Intuitive picture of fp(sgn(x) log jxj=x).............. 55 2.1 For the proof of Lemma 2.4 . 73 2.2 To Lemma 2.8 . 79 2.3 Contour in Example 2.9 . 83 2.4 Contours for Definition 2.22 . ..