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Introduction of the Operator h Through the Convolution Ring C

  • K. Yosida
Chapter
  • 346 Downloads
Part of the Applied Mathematical Sciences book series (AMS, volume 55)

Abstract

The totality of complex-valued continuous functions a(t), b(t), f(t) and so forth defined on the interval [0,∞) will play a particularly important role in the operational calculus; we shall denote the class of those functions by C[0,∞) or simply by the letter C. The convolution of two functions a = a(t) and b = b(t) of C is defined by
$$ (a*b)(t) = a * b(t) = \int_{0}^{t} {a(t - u)b(u)du} (0 \mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle=}\vphantom{_x}}$}} t < \infty ), $$
(1.1)
and we have PROPOSITION 1. a*b belongs to C; i.e., a*b(t) is a continuous function defined on [0, ∞).

Keywords

Continuous Function Complex Number Integration Operator Unit Function Differentiation Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1984

Authors and Affiliations

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

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