Introduction to the Weierstrauss Approximation Theorem // A Proof with Bernstein Polynomials

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I show how to prove the Weierstrauss approximation theorem in this lecture, which states that every continuous function may be approximated by polynomials to any accuracy. We follow a method introduced by Bernstein that gives a constructive proof. This is a complete lecture that gives an introduction to the Weierstrauss theorem. Bernstein's constructive proof is much more accessible than Weiestrauss' original proof and here I presented it to my undergraduate Numerical Analysis students.

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Music:
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0:00 Start
3:04 White Board Start - Weierstrauss Approximation Theorem
10:29 Bernstein Polynomials
14:05 Properties of Bernstein Polynomials
21:20 Weiestrauss Theorem Proof
41:38 A Bernstein Identity
46:59 Closing Remarks

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Thank you! May I ask in which steps of the Weierstrass proof we use the Bernstein properties that you discussed before? This is not clear to me, where do we need the property of it being linear, and the positivity, etc.? I don't see where we apply these to the actual Weierstrass theorem.

pyotrilyichtchaikovsky

Weierstraß that's correct name of this guy
W is read approximately like V
st is read like sht
ß is equvalent to ss but placed incorrecty produces spelling error

holyshit

Just in case somebody else is confused due to the offset in video & audio: download the video and delay the audio by 26.5 seconds to get it in sync with the video.

weinihao

Hello! In your proof you use assumption that f is differentiable. It can be proven without this assumption.

maciess

Great content but this video needs editing. Watching you write out things you say is not a good use of time.

joshstephenson

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