Weierstrass Polynomial Approximation Theorem

[@user-sc2jz9ng6k]

The existence of the Weierstrass function (everywhere continuous but nowhere differentiable) would make for a great video as well.

[@virajmanwadkar6420]

This is an AWESOME presentation! One of the most cogent proofs I have ever seen!!

[@aloysiusgodinho7240]

Absolutely wonderful presentation. Certainly one of the more concise proofs for Weierstrass approximation thm.

[@DrChrisTisdell]

+Aloysius Godinho Thanks and best wishes!

[@omargaber3122]

Wonderful thank you very much

[@matthewjames7513]

Epic proof ! I can't believe I can follow it after so many years of not doing pure maths! Thanks Chris! Is there a Weierstrass polynomial that can approximate periodic functions like a Fourier series can?

[@DrChrisTisdell]

+Matthew James Fourier series are awesome, since they can approximate periodic functions with jumps (discontinuities). I can't see how a Weierstrass polynomial could do that as they rely on continuous functions.

[@kiwisekay3866]

Any function is periodic and continuous can be approximated using a trigonometric polynomial, which is just the second version of Weierstrass' approximation theorem!

[@miles6875]

Love your work

[@CXT14GamerMouse]

First of all, thank you for the awesome presentation. I use it as my primary source for a presentation at university. It is the easiest proof I could find. There are proofs which work almost the same, but with variables as boundaries to the integrals, which looks confusing and you showed me it isn't important. I also want to ask a question: On the slide "Showing the approximation" you say M>0, I think it is also possible to be zero, I know it isn't important for the proof, but still... nice style with the elmo, I quite like to work with the elmo too and wasn't aware of this possibility to use it

[@DrChrisTisdell]

Thanks! Glad you could use it at your university regarding a presentation. Great to see your appreciation for document cameras!

[@George-sc3xf]

Neat, haven't seen one like this before I don't think! Funny how different a feel this has to what I was looking for a proof of, the Stone-Weierstrass version. Thanks!

[@DrChrisTisdell]

+George L Great story!

[@juniorcyans2988]

Weierstrass’s life path is one more example showing that one must do whatever she/he truly loves. Luckily I’m doing what I love, physics!

[@DrChrisTisdell]

Agree! He's one of my heroes. 👍

[@Mahmood42978]

I'll be doing a presentation on the Weierstrass Preparation Theorem next Monday. Hopefully this video gives me some insights.

[@almulakimaalimalriadiat9068]

how did it go ?

[@carlosj3491]

Thanks Dr Chris, this was a very helpful video. I will susbcribe to watch more videos.

[@hanihaddad34]

Great proof. But what i do not understand, is why we define our Jn and Pn(x) just the way you did in this proof. How did you come up with it ?

[@doaam2821]

Thank you we really appreciate your work

[@ilkinond]

Great explanation Dr. Chris - thanks!

[@ML-md1jt]

This is a good laugh.... It is indeed a good try to make it interesting and exciting.

[@DrChrisTisdell]

Mike Lueng "Prove theorem, be happy"! :-)

[@AlqGo]

Why is this amusing?

[@rebeccahardenbrook8414]

Hi, Dr. Tisdell! I was wondering if Landau integrals ever come up in other areas of analysis. This proof is much simpler than ones I have seen; of course, that is in part due to the convenience of the Landau integral. Thank you for this video! I feel much more sure of how Weierstrass' approximation theorem works!

[@DrChrisTisdell]

Hi Rebecca! They probably do come up somewhere in analysis, but I must admit that I've never had to use them. Since there has been a few months in between your post and my reply, I wonder if you found any other examples?

[@عدنانمحمد-ل6ف]

give an applied example about best approximation...thanhs

[@NewCalculus]

Actually there is a much simpler way using Lagrange Interpolation. So simple that even a high school student can understand.

[@DrChrisTisdell]

+John Gabriel I'd love to see that one, too.

[@NewCalculus]

+Dr Chris Tisdell Start at page 2 of the following: drive.google.com/open?id=0B-mOEooW03iLZEVpcHVxclNXZk0 This is an article I wrote on Cotes's integration formula many years ago. It uses the LaGrange polynomial which can be used to approximate any polynomial.

[@waraftab5693]

@@DrChrisTisdell please upload mcq on this topic

[@leonkayombo4316]

Hello Dr Chris. Would you please teach the topic directly. Without having to show present-written notes.

[@peelysl]

Lol math Olympiad prep