82 Bounds on convolution formula

(i) Change variable to $u=y-x$.

(ii) is easy.

(iii) We choose $M,\beta$ such that

$$|f(x)|\le M{e}^{\beta x},|g(x)|\le M{e}^{\beta x}\mathit{\hspace{1em}\hspace{1em}}(x>0);$$

then

$$|f(x-y)g(y)\le M^2{e}^{\beta x}$$

so

$$\left|{\int }_0^{x}f(x-y)g(y)𝑑y\right|\le x{M}^2{e}^{\beta x}$$

so $f\ast g$ satisfies $(E)$.