-2

In the answer of this
convolution is well-defined and differentiable for continuous $f$ and differentiable $g$ with compact support question I didn't understand how Dominated convergence theorem (DCT) is used to interchange limit and integration

Sorry for asking this separately because I am unable to ask in the main answer. Please help me to understand this. Thanks!

Mathqwerty987
  • 9

1 Answers1

1

$\int_K\frac{f(u)(g(u-x)-g(u-x_0))}{x-x_0} \, du$

is an integration over a compact set $K$ and $g,f$ are continuous therefore bounded. So you have a integrable function as a dominator and can use dominated convergence.

cogitoergoboom
  • 313
  • Thank for the answer. I understand that numerator of the integrand is bounded, but I am worried about denominator "$x-x_0$", is $1/(x-x_0)$ bounded as well! – Mathqwerty987 Sep 20 '22 at 20:18
  • integration is over $u$ so $\frac{1}{x-x_0}$ is constant. Do you know how to accept an answer? Please be also aware of https://math.stackexchange.com/tour for further posts. – cogitoergoboom Sep 20 '22 at 20:45