Uploaded October 2023 | Updated September 2026, 1 week ago
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 - math.auckland.ac.nz/hat/people/weierstrass.html
//Code
Bit Bucket - bitbucket.org/joelrosenfeld/weierstrass-approximation
//Books
Rudin - amzn.to/3KeL0IC
Dummit and Foote Abstract Algebra amzn.to/3QxGvwN
Folland Real Analysis amzn.to/3dCuCao
Hungerford Algebra amzn.to/3pIJyGV
Pedersen Analysis NOW amzn.to/3c27MZo
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Real Analysis Survival Guide youtu.be/v5rD0B-zfXw
This is why you have imposter syndrome youtu.be/5CN_KovhHbY
SoME2 Entry youtu.be/3ARBkAM2hRg
Real Analysis:
How to think of Real Numbers youtu.be/fD9VggL7RYU
The Banach Fixed Point Theorem youtu.be/_I790SqNDjs
The Real Analysis Survival Guide youtu.be/v5rD0B-zfXw
Control Theory:
The Analyticity of the Laplace transform youtu.be/FIMkbFQL6XM
Introduction to Control Theory youtu.be/0v4WFmOm764
//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:
epidemicsound.com/referral/644nao
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Canon SL3: amzn.to/3nZ11KU
Canon T6i: amzn.to/3FUpkQh
Rode VideoMic: amzn.to/3lhldGa
Blue Yeti Microphone: amzn.to/3I1y88N
Yeti Nano Microphone: amzn.to/3I1mriA
SanDisc 256GB SD Card: amzn.to/3E3LOOr
Neewer 5600K USB LED Lights: amzn.to/3xvB9cN
Neewer 18 inch Ring Light: amzn.to/2ZvgCsc
Camera Power Adapter: 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
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 - math.auckland.ac.nz/hat/people/weierstrass.html
//Code
Bit Bucket - bitbucket.org/joelrosenfeld/weierstrass-approximation
//Books
Rudin - amzn.to/3KeL0IC
Dummit and Foote Abstract Algebra amzn.to/3QxGvwN
Folland Real Analysis amzn.to/3dCuCao
Hungerford Algebra amzn.to/3pIJyGV
Pedersen Analysis NOW amzn.to/3c27MZo
//Watch Next
Real Analysis Survival Guide youtu.be/v5rD0B-zfXw
This is why you have imposter syndrome youtu.be/5CN_KovhHbY
SoME2 Entry youtu.be/3ARBkAM2hRg
Real Analysis:
How to think of Real Numbers youtu.be/fD9VggL7RYU
The Banach Fixed Point Theorem youtu.be/_I790SqNDjs
The Real Analysis Survival Guide youtu.be/v5rD0B-zfXw
Control Theory:
The Analyticity of the Laplace transform youtu.be/FIMkbFQL6XM
Introduction to Control Theory youtu.be/0v4WFmOm764
//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:
epidemicsound.com/referral/644nao
//Recording Equipment
Canon SL3: amzn.to/3nZ11KU
Canon T6i: amzn.to/3FUpkQh
Rode VideoMic: amzn.to/3lhldGa
Blue Yeti Microphone: amzn.to/3I1y88N
Yeti Nano Microphone: amzn.to/3I1mriA
SanDisc 256GB SD Card: amzn.to/3E3LOOr
Neewer 5600K USB LED Lights: amzn.to/3xvB9cN
Neewer 18 inch Ring Light: amzn.to/2ZvgCsc
Camera Power Adapter: 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
