I have a Convolution integral $$ \int_{t_0}^{t} \int_{t_0}^{\tau} C(t-t')C(\tau -t'') \delta(t''-t') dt'' dt'=\int_{t_0}^{t} C(t-t')C(\tau -t') dt'= ? $$
I do not know how to proceed any further, $\int_{t_0}^{t} C(t-t')C(\tau -t') dt'= ?$ Anyone who can guide me about it? Thanks in advance.
$$C(t)=\int_{-\infty}^{\infty} d\omega e^{i\omega t} f(\omega) J(\omega)$$
\begin{align*} \int_{t_0}^{t} \int_{t_0}^{\tau} C^\sigma(t-t') C^\sigma(\tau -t'') \langle \nu_{it'}\nu_{jt''} \rangle dt'' dt' \\& = \int_{t_0}^{t} \int_{t_0}^{\tau} C^\sigma(t-t') C^\sigma(\tau -t'') \delta_{ij}\delta(t'-t'') dt'' dt' \end{align*} For $i=j, \delta_{ij}=1 $ \begin{align*} = \int_{t_0}^{t} C^\sigma(t-t') C^\sigma (\tau -t') dt' \qquad\qquad \end{align*} Here, $C^\sigma(t) = \int_{-\infty}^{\infty} d\omega f^\sigma(\omega) J(\omega) e^{i\sigma\omega t}$. Solving the integral for $C^\sigma$ \begin{align*} \int_{t_0}^{t} C^{\sigma}(t-t')C^{\sigma}(\tau -t')dt' &= \int_{t_0}^{t} (\int_{-\infty}^{\infty} d\omega f^\sigma(\omega) J(\omega) e^{i\sigma\omega (t-t')})(\int_{-\infty}^{\infty} d\omega' f^\sigma(\omega') J(\omega') e^{i\sigma\omega' (\tau -t')}) dt' \\& = C^\sigma(t)C^{\sigma'}(\tau) \int_{t_0}^{t} e^{-i\sigma(\omega+\omega') t'}dt' \\& = C^\sigma(t)C^{\sigma'}(\tau) \frac{1}{-i\sigma(\omega+\omega')}(e^{-i\sigma(\omega+\omega') t} -e^{-i\sigma(\omega+\omega') t_0} ) \end{align*}