Applied Hyperfunction Theory

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. Front Matter....Pages i-xix Introduction....Pages 1-10 Operations on Hyperfunctions....Pages 11-24 Basic Hyperfunctions....Pages 25-52 Hyperfunctions Depending on Parameters....Pages 53-81 Fourier Transformation....Pages 83-99 Fourier Transformation of Power-Type Hyperfunctions....Pages 101-114 Upper (Lower)-Type Hyperfunctions....Pages 115-131 Fourier Transforms-Existence and Regularity....Pages 133-145 Fourier Transform-Asymptotic Behaviour....Pages 147-164 Periodic Hyperfunctions and Fourier Series Fourier Series....Pages 165-204 Analytic Continuation and Projection of Hyperfunctions....Pages 205-223 Product of Hyperfunctions....Pages 225-247 Convolution of Hyperfunctions....Pages 249-282 Convolution of Periodic Hyperfunctions....Pages 283-301 Hilbert Transforms and Conjugate Hyperfunctions....Pages 303-326 Poisson-Schwarz Integral Formulae....Pages 327-355 Integral Equations....Pages 357-380 Laplace Transforms....Pages 381-391 Epilogue....Pages 393-394 Back Matter....Pages 395-433