# Applied Hyperfunction Theory

Springer Science & Business Media, May 31, 1992 - Mathematics - 438 pages
Generalized functions are now widely recognized as important mathematical tools for engineers and physicists. But they are considered to be inaccessible for non-specialists. To remedy this situation, this book gives an intelligible exposition of generalized functions based on Sato's hyperfunction, which is essentially the `boundary value of analytic functions'. An intuitive image -- hyperfunction = vortex layer -- is adopted, and only an elementary knowledge of complex function theory is assumed. The treatment is entirely self-contained.
The first part of the book gives a detailed account of fundamental operations such as the four arithmetical operations applicable to hyperfunctions, namely differentiation, integration, and convolution, as well as Fourier transform. Fourier series are seen to be nothing but periodic hyperfunctions. In the second part, based on the general theory, the Hilbert transform and Poisson-Schwarz integral formula are treated and their application to integral equations is studied. A great number of formulas obtained in the course of treatment are summarized as tables in the appendix. In particular, those concerning convolution, the Hilbert transform and Fourier transform contain much new material.
For mathematicians, mathematical physicists and engineers whose work involves generalized functions.

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### Contents

 INTRODUCTION 1 2 Satos hyperfunction 2 3 Aim 3 4 Complex velocity and analytic function 4 5 Distribution of vortices and hyperfunctions 6 6 Ordinary functions and hyperfunctions 8 OPERATIONS ON HYPERFUNCTIONS 11 2 Linear combinations 13
 6 Convolution of two periodic hyperfunctions 299 3 Properties of conjugate hyperfunctions 307 6 Formulae of Hilbert transforms 314 7 Standardtype generating function 322 8 Hilbert type transforms 324 9 Summary 326 POISSONSCHWARZ INTEGRAL FORMULAE 327 3 PoissonSchwarz integral formula for a circle 329

 3 Product of a hyperfunction and an analytic function 14 4 Reinterpretation of ordinary functions as hyperfunctions 15 5 Differentiation of hyperfunctions 19 6 Definite integrals of hyperfunctions 22 7 Summary 23 viii 25 10 Hz lz sgnz 40 HYPERFUNCTIONS DEPENDING ON PARAMETERS 53 8 Powertype hyperfunctions 67 10 Calculation of pf integrals 75 11 Summary 81 3 Theorems about Fourier transformation 88 5 Examples of calculations of Fourier transforms 97 2 Tza 103 6 FxmlogxnHx xmlogxn xmlogxnsgnx 109 6 Upper lower powertype hyperfunctions and their Fourier trans 126 4 Regularity of GQ Ftzl+z on the axis 137 8 Sufficient conditions for the existence of convolution 267 ill Fourier transforms of convolutions 275 15 Summary 282 4 Convolution of periodic hyperfunctions 293
 4 Generalization of the PoissonSchwarz integral formula 331 5 RiemannHilbert problem 337 6 Integral equations related to Hilbert transforms 339 7 IZo ftt x1 dt gx oo x oo 340 9 faftt x1 dt gx ax b 343 10 HJx axfx + 0x 348 11 i ftlogt xdt gx 1 x 1 352 12 Summary 355 INTEGRAL EQUATIONS 357 3 Solution of convolution equations 358 4 Examples of application 360 5 Alternative method 362 6 Volterra integral equations 363 7 Abels integral equation 365 8 JJWe1dt gx x0 368 LAPLACE TRANSFORMS 381 REFERENCES 395 Appendix E Uppertype and lowertype hyperfunctions 409 Fourier transforms 427 xvi 435 Copyright