Error in approximation by convolution with a mollifier or a Gaussian

Question

Let $f : \mathbb{R} \to \mathbb{R}$ be bounded and supported on some finite non-empty open interval $(a,b)$. For $\delta > 0$, let $\phi_\delta(x) = \delta^{-1}\phi(x/\delta)$ be the standard mollifier. If $f$ is smooth, what is an upper bound for $\|(f \ast \phi_\delta) - f\|_{L_p}$ in terms of $\delta$? I'm especially interested in the case when $p=1$. What in the case when $f$ is only continuous? It would be interesting to know what happens when $\phi(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ is the Gaussian as well. I think these things should be well known, but I was not able to find a reference online. Thanks!