Applied Hyperfunction Theory

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Springer Science & Business Media, May 31, 1992 - Mathematics - 438 pages
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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
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