Proof and Intuition for the Weierstrass Approximation Theorem

[@germangonzalez3063]

The way you teach is impressive. I have been around understanding all this and finally came into your channel.

[@JoelRosenfeld]

I’m glad you like it! It takes a lot of effort to put this all together. Happy to have you here!

[@zoedesvl4131]

I remember my professor of functional analysis II (equivalent to part III of Rudin's Functional Analysis) proved this theorem because this theorem is a crucial part of a theorem in operator theories.

[@JoelRosenfeld]

Absolutely, I honestly think it's one of the most crucial theorems out of real analysis. It's used all over the place, including dynamical systems theory, approximation theory, operator theory, etc. I got excited about it, so I jumped the gun a bit and just made a video on the topic. I really should build up concepts of uniform convergence and continuity first. As well as integration theory. That is, if I want to keep the channel self contained. haha

[@paul_tee]

wow, this is probably the most high-quality content i've seen that's aimed at undergraduates. can you give some references to moments of a function? in particular, i'm wondering if you can phrase this idea on a riemannian manifold. there's something that came up in my area that looks a bit like the first moment.

[@JoelRosenfeld]

I'm happy you like the video. It took about two weeks of my spare time to throw it together. I try to keep the quality up, but that takes a lot of time! Moments and moment problems are something that come up a bunch in probability and measure theory. If you look up information on the "Hamburger Moment problem," that road should lead you to a bunch of references related to analysis. I don't have any particular reference to give you outside of that, though.

[@matejrajchl3596]

Very cool video! I really enjoy the format where you actually write on the whiteboard, really helps me to process the information in correct sequence. Thank you very much for your informative videos!

[@JoelRosenfeld]

I'm really happy you like!

[@patagonia12312]

maravilloso!!

[@mmariokart231]

Hey! Awesome!! Thank you for making this!!

[@surajpowar4915]

Really interesting one. I have a suggestion, we would really love to hear how you build your computational skills as a graduate student or what advice would you have for a graduate student to improve their coding skills in terms of maths. If it is out of your scope of your channel, I understand. As always great video. ❤️❤️

[@JoelRosenfeld]

I actually was planning a video like that to come soon. I’m still in the scripting phase. Is there anything you’d like me to talk about in particular? What would you like to use coding skills for?

[@surajpowar4915]

@@JoelRosenfeld Thank you for your kind response. I am a graduate student and I am hoping to get a PhD in computational mathematics focusing in the area of numerical analysis and simulations. I see the scope of coding to simulate numerical results or solve ode or ode equations analytically and then simulate it on python or matlab. But not sure how to go about it.

[@NR-ft6cj]

The Pilot Board Master marker! Love them!!!

[@JoelRosenfeld]

If I have to use a whiteboard marker, might as well use the best!

[@NR-ft6cj]

@@JoelRosenfeld YES

[@okoyoso]

The form of that integral with the term |f(x+t)-f(x)|dt looks like you could apply the mean value theorem. Although that might create a dependence on x?

[@JoelRosenfeld]

If we were allowing f to be differentiable then we probably would use it. Here we are only allowing continuous functions

[@okoyoso]

@@JoelRosenfeld Makes sense, cool

[@brendanchamberlain9388]

Thank you! Can you do a video on liminf/limsup? Rudin uses them in thm3.31 to prove the limit formula for e, but his proof confuses me because I feel I do not fully understand liminf and limsup at this point 😢 If you don’t want to do a video do you know of any good resources that nicely cover liminf and limsup?

[@JoelRosenfeld]

Sure, I'll give it some thought. I was planning on throwing together a "problem video" where I do a bunch of exercises from the book. Anything else you'd like on the list?

[@brendanchamberlain9388]

@@JoelRosenfeld maybe exercise 2.29?

[@JoelRosenfeld]

That’s a good one! lol I will have to wrack my brain to remember how I did it years ago

[@strikeemblem2886]

​@@JoelRosenfeld Ex 2.29 is asked and answered in stackexchange. There is even a thread to collect different ways to prove it.

[@JoelRosenfeld]

@@strikeemblem2886 thanks! I figured out a way to do it this afternoon. Essentially, you can set up an equivalence class of points that are in common intervals. Countability follows by finding a rational point in each set.

[@ktyoung1004]

❤❤❤❤❤