Young’s Convolution Theorem

Let 1\leq p,q\leq \infty and 1/p+1/q\geq 1, and let 1/r=1/p+1/q-1. If f\in L^p(\mathbb{R}^n) and g\in L^q(\mathbb{R}^n), then f*g\in L^r(\mathbb{R}^n) and \displaystyle \|f*g\|_r\leq\|f\|_p\|g\|_q.

Amazing Theorem! If q=1, then \|f*g\|_p\leq\|f\|_p\|g\|_1.

Advertisements

About mathtuition88

http://mathtuition88.com
This entry was posted in math and tagged , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s