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Proof and Intuition for the Weierstrass Approximation Theorem

This is an in depth look at the Weierstrass Approximation Theorem and the proof that can be found in Rudin's Principles of Mathematical Analysis. This proof resolves the polynomial approximation problem, a moment problem, and can even be seen as previewing Dynamic Mode Decompositions. //Articles Anton Schep - Weierstrass' Proof of the Weiestrass Approximation Theorem https://people.math.sc.edu/schep/weie... Weierstrass' 1885 Manuscript - https://www.math.auckland.ac.nz/hat/p... //Code Bit Bucket - https://bitbucket.org/joelrosenfeld/w... //Books Rudin - https://amzn.to/3KeL0IC Dummit and Foote Abstract Algebra https://amzn.to/3QxGvwN Folland Real Analysis https://amzn.to/3dCuCao Hungerford Algebra https://amzn.to/3pIJyGV Pedersen Analysis NOW https://amzn.to/3c27MZo //Watch Next Real Analysis Survival Guide    • The Real Analysis Survival Guide   This is why you have imposter syndrome    • This is WHY you have Imposter Syndrome   SoME2 Entry    • The Mathematics of Radio Transmissions   Real Analysis: How to think of Real Numbers    • Intro to Real Analysis: How to think ...   The Banach Fixed Point Theorem    • Fixed Point Iteration - Loki SLAPPED ...   The Real Analysis Survival Guide    • The Real Analysis Survival Guide   Control Theory: The Analyticity of the Laplace transform    • Morera's Theorem in Practice - Laplac...   Introduction to Control Theory    • Introduction to Control Theory   //Music Most music provided by Epidemic Sound with the exception of one song by Tom Fox at Johnny Harris' Patreon Use this referral link to get a 30 day free trial with Epidemic Sound for your YouTube channel: https://www.epidemicsound.com/referra... //Recording Equipment Canon SL3: https://amzn.to/3nZ11KU Canon T6i: https://amzn.to/3FUpkQh Rode VideoMic: https://amzn.to/3lhldGa Blue Yeti Microphone: https://amzn.to/3I1y88N Yeti Nano Microphone: https://amzn.to/3I1mriA SanDisc 256GB SD Card: https://amzn.to/3E3LOOr Neewer 5600K USB LED Lights: https://amzn.to/3xvB9cN Neewer 18 inch Ring Light: https://amzn.to/2ZvgCsc Camera Power Adapter: https://amzn.to/3D3upUu DISCLAIMER: The links above in this description may be affiliate links. If you make a purchase with the links provided I may receive a small commission, but with no additional charge to you :) Thank you for supporting my channel so that I can continue to produce mathematics content for you! 0:00 The Weierstrass Approximation Theorem 2:14 First Simplification 4:42 Uniform Convergence 7:30 Can never be too old to do math! 8:18 The Main Characters of the Proof 10:43 Walter Rudin's Approach 14:11 Qn - A Delta Sequence 18:30 Uniform Continuity 19:27 The Proof of the Weierstrass Approximation Theorem 25:17 MATLAB Code for the Weierstrass Approximation Theorem 26:11 Is it a Polynomial? 28:27 Closing Remarks