Introduction of the Operator h Through the Convolution Ring C

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 1.1$$ (a*b)(t) = a * b(t) = \int_{0}^{t} {a(t - u)b(u)du} (0 \mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle=}\vphantom{_x}}$}} t < \infty ), $$ and we have PROPOSITION 1. a*b belongs to C; i.e., a*b(t) is a continuous function defined on [0, ∞).