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Home , Hilbert transform, Hyperfunction, Weierstrass transform

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1 2 Introduction to 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 ...... 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 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 , 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 of Hyperfunctions ...... 137 2.7.1 Definition and Existence of the ...... 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 ...... 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 ...... 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 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 ∈ 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 ...... 375 A.2.1 Functions of Exponential Type ...... 375 A.2.2 Laplace Hyperfunctions and their Transforms ...... 384 A.3 Some Basic Theorems of 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 of the Dirac impulse . . . 19 1.5 Intuitive picture of the second derivative of the Dirac impulse . . 21 1.6 Piecewise real ...... 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, ∞)...... 42 1.13 Integration loop (−∞, as)...... 42 1.14 Intuitive picture of fp(sgn(x) log |x|/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 ...... 95 2.5 Contours of Example 2.20 ...... 97 2.6 Contours in the proof of Proposition 2.28 ...... 102 2.7 Intuitive picture of the hyperfunction of Example 2.26 ...... 111 2.8 circles: singularities of g(x), filled disks: singularities of f(x).. 120 2.9 Closed contour γ in Proposition 2.44 ...... 122 2.10 Closed contour γ for three groups of singularities ...... 123 2.11 Closed contour in Example 2.35 ...... 124 2.12 Contour Cb ...... 126 2.13 Contours in the ζ-plane ...... 137 2.14 Contours in the θ-plane ...... 140

3.1 Contour `(t0, α, Ω) ...... 156 3.2 Contours in Proposition 3.5 ...... 168 3.3 Roots and partition of C in Example 3.19 ...... 196 3.4 Integration loop Cz ...... 212 3.5 Integration loop Cz deformed ...... 212 3.6 Contours for the proof of Proposition 3.22 ...... 215 3.7 Contour (c, cz)...... 216 3.8 Contours in the proof of (3.152) ...... 221

vii 5.1 Contour in the proof of Proposition 5.6 ...... 281

7.1 Schl¨afli’scontour S ...... 340 7.2 Hankel contours H1 and H2 ...... 341 7.3 Contour (, λ)...... 354 7.4 Contours in the proof of Proposition 7.9 ...... 360

A.1 P´olya field of the doublet δ(x + 1/2) − δ(x − 1/2) ...... 374 A.2 z-plane ...... 379 A.3 s-plane ...... 379 A.4 Sets Σ(α, β),Z(α, β), contour ΓR ...... 381 A.5 Contour γR in the case of a sector S(0, 2π)...... 383

viii Preface

This textbook is about generalized functions and some of their integral trans- forms in one variable. It is based on the approach introduced by the Japanese mathematician . This is to be mentioned because the term hyper- function that Sato has given to his generalization of the concept of function, is sometimes used today to denote generalized functions based on other ap- proaches (distributions, Mikusinski’s operators etc.). I have written this book, because I am delighted by the intuitive idea behind Sato’s approach which uses the classical complex function theory to generalize the notion of function of a real variable. To my opinion, Sato’s way to introduce the generalized concept of a function is less abstract than the one of who defines his distributions as linear functionals on some space of test functions. On the other hand, I was quickly led to recognize that very few colleagues (mathematicians included) knew something about Satos’s approach. Perhaps Sato and his school is not innocent for this state of affair. During several decades no elementary textbook addressing a wider audience was available (at least in English). Zealots delighted by the appealing intuitive idea of the approach have probably been rather rejected by the adopted style of exposition and the highly ambitious ab- stract mathematical concepts used in those few books. Fortunately, some years ago, I found Isac Imai’s Book Applied Hyperfunction Theory which explains and applies Sato’s hyperfunctions in a concrete, but nontrivial way, and thereby reveals their computational power. Without the help of Imai’s book, I would have been definitely repelled too by the available sophisticated texts as per- haps many others before me. So, for the writing of my book I am indebted to Imai, mainly for the first chapter, parts of the second and entirely for the fifth chapter. The objective of my book is to offer an introduction to the theory of hyperfunctions and some of their integral transforms that should be readable to a wider audience (applied mathematicians, physicists, engineers) than to just some few specialists. The prerequisites are some basics notions on complex function theory of one variable and of the classical Laplace and Fourier trans- formation. Since I am no friend of theories for their own sake, I have inserted throughout the whole book some simple applications mainly to the theory of integral equations. Chapter 1 is an elementary introduction to generalized functions by the hyperfunction approach of Sato. With a few basic facts on complex function theory the basic lines of a theory of generalized functions is presented that should be easy to read and easy to be understood. It shows the basic intuitive concept of a hyperfunction of one variable. Chapter 2 discusses the analytic properties of hyperfunctions. Here, the spe- cific methods of complex analysis come into bearing. Since a hyperfunction is

ix defined by specifying a defining or generating function, we treat in some detail the question on how to construct a defining function such that the correspond- ing hyperfunction interprets the given ordinary function (problem of embedding an ordinary function in the set of hyperfunctions). We shall see that the an- swer to this problem is not always unique. A hyperfunction which interprets a given ordinary function on a specified interval is said to be a projection of this function to the interval. Given a hyperfunction on an interval (a0, b0), the anal- ogous problem of finding another hyperfunction that equals the specified one on a smaller interval (a, b) ⊂ (a0, b0) and vanishes outside (a, b) is then treated. This leads to the notion of the so-called standard defining function, a term first defined for hyperfunctions with a compact support, then extended to hyper- functions defined on an infinite interval. Imai extends the notion of a standard defining function in another way than Sato has done when the hyperfunction is not perfect, i.e. has a non-compact support. Because Imai’s extension is found to be useful for the discussion of Hilbert transformation, I shall use the term ”strong defining function” to avoid confusion of the two notions. An introduc- tion to periodic hyperfunctions and their Fourier series then follows. The last theoretical part of this chapter discusses convolutions of hyperfunctions. Some informal remarks on integral equations conclude the chapter. Chapter 3 treats the Laplace transform of hyperfunctions. It is somewhat the main axis of the part about integral transformations. While other texts about generalized functions often treat Fourier transforms in the first place and then used the established Fourier transformation to define the Laplace transfor- mation, I will do it in the converse way. I introduce the Laplace transform of a hyperfunction by using a loop integral of the Hankel type over the defining func- tion. Since simplicity of the presentation, together with many concrete exam- ples without digging into finer mathematical points in the arguments, has been aimed, this more elementary approach in the main text is presented rather than Komatsu’s theory of Laplace hyperfunctions. But since the Laplace transforma- tion is a central subject of the book, I have presented an outline of Komatsu’s approach in the Appendix B. Chapter 4 is about Fourier transforms of hyperfunctions. Fourier transfor- mation is without doubt the greatest beneficiary of the theory of generalized functions. Generally the Fourier transformation is treated in the first place in most approaches to generalized functions. From a mathematico-logical point of view this may be justified, however, from a computational and applied stand- point Laplace transformation is often more appropriate. I define the of a hyperfunction by using the already established Laplace trans- form. This approach has the advantage that the available extended tables of Laplace transforms can be used. Chapter 5 treats Hilbert transforms of hyperfunctions. In this chapter I mainly follow Imai. The concept of a strong defining function plays an important role here because the existence of the of a hyperfunction is intimately connected to the existence of a strong defining function. A section on analytic signals can also be found there. For the last two chapters about Mellin and Hankel transformations I had to be entirely self-supporting. While there is an abundant literature about clas- sical Mellin and Hankel transformation of ordinary functions, I could not find anything at least in English, German or French on Mellin or Hankel transfor- mation of hyperfunctions. A. H. Zemanian [40] treats the Mellin and Hankel

x transformation of generalized functions based on Laurent Schwartz’s theory of distributions. I have finally succeeded to convey the Mellin transformation to hyperfunctions by taking advantage that the Mellin transformation is in some sense a reformulation of the two-sided Laplace transformation. A simple change of variables allows passing from the Laplace to the Mellin transformation, and vice versa. Because the Laplace transformation of hyperfunction has been firmly established in Chapter 3, it was finally straightforward to establish the Mellin transformation of hyperfunctions by exploiting this connection to the hilt. It was harder to define the Hankel transform of a hyperfunction. I have even- tually found a way by working on the line of Mac Robert’s proof of the classical Hankel transformation which uses the so-called Lommel integrals of Bessel func- tions. The established theory about the generalized Hankel transformation then finally works for hyperfunctions of slow growth. Throughout the whole book a particular function and its hyperfunction coun- terpart comes up again and again: the Heaviside function Y (x) and the unit- step hyperfunction u(x). For didactic reasons, and after some hesitations, I have made a distinction between them. This may be pedantic and I agree that it is not absolutely necessary because there is no great danger of confusion by the use of a unified notation for the two. Thus, the reader should feel free to replace everywhere Y (x) by u(x). Systematically, all contours in the complex plane will be positively directed unless noted to the contrary. This will produce minus signs before integrals where some readers will not expect them, perhaps. The main tool used in this book consists of contour integration in the com- plex plane. You will find numerous integrals taken on a closed contour or on an infinite loop. We liberally interchange in many places the order of integration in multiple integrals or the integral and an infinite series. In order to keep the flow of the arguments fluid, I do not generally justify these steps in detail. How- ever, at the disposition for readers interested in such technical details, I have collected in the Appendix A3 the principal theorems generally used to justify such interchanges of limit operations. Lastly, a few remarks about what you cannot find in this book and about what I do not have any pretensions. The theory goes not very deep. No sheafs and other sophisticated concepts are mentioned. The intended message is rather conveyed through many concrete examples. Also, and this is certainly a short- coming, no hyperfunctions of several variables are treated.

La Neuveville, Switzerland, October 2009, U.G.

xi xii 406 Index

B-transform, 205 differentiation of piecewise real analytic functions, 22 analytic function Dirac comb, 68, 70, 130, 151, 268, 278 analytic continuation of, 93 Dirac impulse hyperfunction, 7 identity theorem, 93 distribution, 2 Schwarz’s Reflection Principle, 93 , 293 equivalent defining function, 4 Euler’s differential equation, 327 Bessel functions exponential integral of the first kind, 338 En(z), 177 of the second kind, 342 Ei(x), 179 of the third kind, 341 E1(z), 177 biorthogonal system of hyperfunctions, complementary, 178 39 complementary*, 180 canonical splitting, 189, 250 Fourier series of a periodic hyperfunc- Cauchy-type integral, 71 tion, 132 compact convergence, 63 Fourier series of the Dirac comb, 70 complex neighborhood, 3 Fourier transform conjugate Fourier series, 295 as a holomorphic hyperfunction, conjugate , 86 241 convergence factor of the first kind, 87 computed via Laplace transforma- convergence factor of the second kind, tion, 243 90 convolution property, 259 convergence in the sense of hyperfunc- of hyperfunctions with compact sup- tion, 64 port, 241 convex hull of singularities, 380 of right-sided or left-sided hyper- convolution algebra, 148 functions, 242 convolution of two hyperfunctions, 138 Reciprocity Rule, 252 convolutions and Laplace transforms, Fourier transform, inverse, 251 216 fractional derivatives, 222 cylinder functions, 338 fractional integrals, 222 cylinder functions, recurrence relations, fractional ordinary differential equation, 343 225 full neighborhood, 91 Dawson’s integral, 264 functions of exponential type, 376 decomposition theorem, 99 defining function, 13 general sign-hyperfunction, 45 delta function, 1 general substitution, 16 derivatives in the sense of hyperfunc- generalized delta-hyperfunction, 29 tions, 18 generalized derivatives, 18

410 Green’s function, 197 physical interpretation, 375 growth index, 162 primitive of, 198 pure imaginary, 28 H¨oldercondition, 77 real part of a, 28 Hadamard’s finite part, 126 singular spectrum of a , 111 half-plane functions (upper and lower), singular support of, 111 74 standard, 89 Hankel functions, 341 support of a, 111 Hankel transform hyperfunctions by using Laplace transformation, analytic continuation of, 113 347 depending on a parameter, 44 different definitions, 337 even / odd, 26 of a hyperfunction, 355 Heisenberg’s, 35 Parseval’s relation, 350 identity theorem, 112 Hankel transforms integral power, 46 Mac Robert’s proof of the inver- non-integral power , 49 sion formula, 346 of slow growth, 241 heat equation, 270 product of, 23, 115 Heaviside function, 1 product of lower and upper, 114 Hermite’s function, 170, 241 real, 28 Hermite’s , 170 real type, 282 Hilbert transform subclass B1(R), 277 and Fourier transform, 290 upper and lower, 56, 280 and integral equations, 297 with finite moments, 284 and strong defining function, 275 with logarithms, 52 classical definition, 273 generalized definition, 274 indicator function, 376 iterated, 278 integral equation of periodic hyperfunctions, 296 Abel’s, 229 of upper and lower hyperfunctions, Fredholm, 151 282 of Cauchy’s type, 268 holomorphic functions and P´olya fields, of convolution type, 152 374 over infinite range, 231 Hurwitz’s zeta function, 330 singular, 152 hyperfunction Volterra, 151 characteristic, 45 integral equations complex-conjugate , 28 Volterra, 226 concentrated at one point, 62 integral of a hyperfunction, 38 defined by one global analytic func- integral over a product of hyperfunc- tion, 94 tions, 121 definition, 4 differentiation of, 18 reciprocal, 323 finite part, 33 integrals with respect to a parameter, holomorphic, 91 44 homologous, 89 interpretation of an ordinary function imaginary part of a, 28 as a hyperfunction, 6 meromorphic, 40, 92 intuitive picture of a hyperfunction, 8 of bounded exponential growth, 165 Jordan’s lemma, 75, 161 perfect, 100 periodic, 128 Laplace hyperfunctions, 214, 384

411 Laplace transform sequential approach, 2 arbitrary support, 168 shift operator as differential operator, inversion formula, 386 25 of a holomorphic function, 377 sign-hyperfunction, 10 of a Laplace hyperfunction, 385 singular point, 6 of a left-sided original, 164 Sokhotski formulas, 81 of a right-sided original, 163 standard defining function, 100, 104 Pincherle’s Theorem, 384 strong defining function, 105 Laplace transforms of a right-sided Dirac comb, 172 uniform convergence in the interior of, right-sided, 223 63 left-sided original, 164 unit-step function, 1 Legendre polynomial, 102 unit-step hyperfunction, 8 linear substitution, 13 upper component, 3 Lipschitz condition, 77 upper half-neighborhood, 3 Lommel’s integral, 345 upper half-plane function, 281 lower component, 3 lower half-neighborhood, 3 Weierstrass transform, 271 Wiener-Hopf Equations, 233 Mellin ∗convolution, 321 Wiener-Hopf technique, 233 Mellin transform ∗convolution rule, 322 zero hyperfunction, 11 ◦convolution, 322 classical, 307 inversion formula, 308, 319 of a product, 325 of hyperfunctions, 311 Parseval’s formula, 325 microanalytic from above, 110 microanalytic from below, 110 ordinary derivatives, 18 ordinary function, 5, 92

P´olya field, 372 Pochhammer’s symbol, 48 Poisson’s summation formula, 266 product rule, 20 projection of a hyperfunction, 94 projection of an ordinary function, 76 pv-convolution, 149 real analytic function, 11 real neighborhood, 91 regular point, 6 Riemann’s zeta function, 330 right-sided original, 162

Sato Mikio, 2 scalar product of hyperfunctions, 39 Schwartz Laurent, 2

412