Convex Norms and Unique Best Approximations
In this video, we explore what it means for a norm to be convex. In particular we will look at how convex norms lead to unique best approximations.
For example, for any continuous function there will be a unique polynomial which gives the best approximation over a given interval.
Chapters
0:14 - Geometry of the Lp Norm
0:45 - Convexity of the Lp Norm
2:13 - Best Approximations are unique for convex norms (proof)
4:41 - Example
The product links below are Amazon affiliate links. If you buy certain products on Amazon soon after clicking them, I may receive a commission. The price is the same for you, but it does help to support the channel :-)
The Approximation Theory series is based on the book "Approximation Theory and Methods" by M.J.D. Powell:
https://amzn.to/4fd13aM
Errata and Clarifications:
The L1/2 "norm" isn't actually a norm! Lp for p less than 1 fail the triangle inequality.
All norms and normed linear spaces are convex, the important bit for the proof is that A must be a convex subset.
In the last section I show a graph labelled exp(-x) but that's not correct looking at the curve. It doesn't hurt the explanation, any curve would've been fine.
This video was made using:
Animation - Apple Keynote
Editing - DaVinci Resolve
Mic - Shure SM58 (with Behringer U-PHORIA UM2 audio interface)
Supporting the Channel.
If you would like to support me in making free mathematics tutorials then you can make a small donation over at
https://www.buymeacoffee.com/DrWillWood
Thank you so much, I hope you find the content useful.
Watch video Convex Norms and Unique Best Approximations online without registration, duration hours minute second in high quality. This video was added by user Dr. Will Wood 26 March 2021, don't forget to share it with your friends and acquaintances, it has been viewed on our site 12,082 once and liked it 485 people.