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/weierstrass.pdf
Weierstrass' 1885 Manuscript - https://www.math.auckland.ac.nz/hat/people/weierstrass.html
//Code
Bit Bucket - https://bitbucket.org/joelrosenfeld/weierstrass-approximation/
//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 https://youtu.be/v5rD0B-zfXw
This is why you have imposter syndrome https://youtu.be/5CN_KovhHbY
SoME2 Entry https://youtu.be/3ARBkAM2hRg
Real Analysis:
How to think of Real Numbers https://youtu.be/fD9VggL7RYU
The Banach Fixed Point Theorem https://youtu.be/_I790SqNDjs
The Real Analysis Survival Guide https://youtu.be/v5rD0B-zfXw
Control Theory:
The Analyticity of the Laplace transform https://youtu.be/FIMkbFQL6XM
Introduction to Control Theory https://youtu.be/0v4WFmOm764
//Music
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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