Operational Calculus

A Theory of Hyperfunctions

• K. Yosida Book

Part of the Applied Mathematical Sciences book series (AMS, volume 55)

1. Front Matter
Pages i-x
2. Integration Operator h and Differentiation Operator s (Classes of Hyperfunctions: C and CH)

1. K. Yosida
Pages 1-4
2. K. Yosida
Pages 5-13
3. K. Yosida
Pages 14-31
4. K. Yosida
Pages 32-38
5. K. Yosida
Pages 39-46
3. Linear Ordinary Differential Equations with Linear Coefficients (The Class C/C of Hyperfunctions)

1. K. Yosida
Pages 47-52
2. K. Yosida
Pages 53-73
4. Shift Operator exp(−λs) and Diffusion Operator exp(−λs1/2)

1. K. Yosida
Pages 74-105
5. Applications to Partial Differential Equations

1. Front Matter
Pages 106-107
2. K. Yosida
Pages 108-123
3. K. Yosida
Pages 124-144
4. K. Yosida
Pages 145-156
6. Back Matter
Pages 157-171

Introduction

In the end of the last century, Oliver Heaviside inaugurated an operational calculus in connection with his researches in electromagnetic theory. In his operational calculus, the operator of differentiation was denoted by the symbol "p". The explanation of this operator p as given by him was difficult to understand and to use, and the range of the valid­ ity of his calculus remains unclear still now, although it was widely noticed that his calculus gives correct results in general. In the 1930s, Gustav Doetsch and many other mathematicians began to strive for the mathematical foundation of Heaviside's operational calculus by virtue of the Laplace transform -pt e f(t)dt. ( However, the use of such integrals naturally confronts restrictions con­ cerning the growth behavior of the numerical function f(t) as t ~ ~. At about the midcentury, Jan Mikusinski invented the theory of con­ volution quotients, based upon the Titchmarsh convolution theorem: If f(t) and get) are continuous functions defined on [O,~) such that the convolution f~ f(t-u)g(u)du =0, then either f(t) =0 or get) =0 must hold. The convolution quotients include the operator of differentiation "s" and related operators. Mikusinski's operational calculus gives a satisfactory basis of Heaviside's operational calculus; it can be applied successfully to linear ordinary differential equations with constant coefficients as well as to the telegraph equation which includes both the wave and heat equa­ tions with constant coefficients.

Keywords

Derivative Finite Hyperfunktion Identity Operatorenrechnung algebra calculus differential equation equation function logarithm proof theorem

Authors and affiliations

• K. Yosida
• 1
1. 1.Kamakura 247Japan

Bibliographic information

• DOI https://doi.org/10.1007/978-1-4612-1118-1
• Copyright Information Springer-Verlag New York, Inc. 1984
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-0-387-96047-0
• Online ISBN 978-1-4612-1118-1
• Series Print ISSN 0066-5452
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