Difference between revisions of "MA598R 7.6" - Rhea

Latest revision as of 04:53, 11 June 2013

Problem #7.6, MA598R, Summer 2009, Weigel

Sneak through the inky black night back to The_Ninja's_Solutions

Suppose $ f \in L^{1}(\mathbb{R}) $ satisfies $ f*f=f $. Show that $ f=0 $.

$ \hat{f} = \widehat{f*f} = \hat{f}^2 $ by problem 2.

Now $ (\hat{f}(\xi))(\hat{f}(\xi)-1)=0 $ so $ \hat{f} = \chi_{A} $ for some set $ A $

But problem 5 gives $ \hat{f} $ is continuous and the limit is zero, hence $ \hat{f}\equiv 0 $

Applying an inverse fourier transfom gives $ f = 0 $ a.e.

$ f = f*f = \int_{\mathbb{R}} f(x-y)f(y)dy = 0 $ because the integral of something that is zero a.e. is zero.

~Ben Bartle

Back to Ninja Solutions

Back to Assignment 7

Back to MA598R Summer 2009