Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm having difficulty with the following problem:

Let $f:\mathbb{R}\to\mathbb{R}$ be continuous on $\mathbb{R}$ such that $\mbox{Supp}\left(f\right)$ is compact and let $g:\mathbb{R}\to\mathbb{R}$ be continuously differentiable. I need to show that the convolution $f*g$ is also continuously differentiable and that the derivative of $f*g$ is $f*g^{'}$.

I am quite stumped.

share|improve this question
add comment

2 Answers 2

up vote 1 down vote accepted

Write the differential quotient for $f * g$: you get two integrals. Change variables in an integral so that you can put the integrals together and collect $f$ inside the integral. See the incremental quotient of $g$ appear inside the integral. Pass to the limit.

share|improve this answer
add comment

For the first part, both your functions are summabel $L_1(\Bbb R)$.

Now have a look at http://people.math.gatech.edu/~heil/6338/summer08/section4c_convolve.pdf

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.