Partial derivative of convolution

Question

I have a convolution:

$$g(x,\alpha) = \int_D \phi(t)f(x-t,\alpha)dt,$$

where $D$ is compact. I need to calculate $\frac{\partial}{\partial \alpha}g(x,\alpha)$. Under what conditions:

$$\frac{\partial}{\partial \alpha}g(x,\alpha) = \int_D \phi(t)\frac{\partial}{\partial \alpha}f(x-t,\alpha)dt\,?$$

Do I need to walk the tortuous road of finding an integrable upper bound for the partial derivative of $f$ in order to appeal to the dominated convergence theorem?

Answer 1

• I found a reference for this. Basically, we need that the partial derivative of $f$ with respect to $\alpha$ is locally integrable with respect to $\alpha$ over any compact set on ${\mathbb R}$:

http://planetmath.org/differentiationundertheintegralsign

It would be desirable to find a more serious reference. I would accept any answer pointing in this direction.

• Another reference:

http://landonkavlie.wordpress.com/