Chebyshev Polynomials | Theory & Practice

Today I discuss what are Chebyshev polynomials and why we need them. After Introducing Chebyshev polynomials and their properties, I will write three separate codes to generate Chebyshev polynomials in MATLAB.
Chebyshev nodes can help us interpolate functions. They are much better than linearly spaced nodes. In the video I explain why Chebyshev nodes are a better option, and provide an example to show that equally spaced nodes may lead to a wrong result.
Watch the video and tell me what you think in the comments.
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Link to other tutorials mentioned in this video:
Proof for Chebyshev Polynomials recursion relation:
• Quintic Equation From ...
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0:00 - Introduction to Chebyshev Polynomials
00:40 - Chebyshev Polynomials in MATLAB
02:40 - Using Chebyshev Polynomials to Interpolate Functions
04:48 - Chebyshev Interpolation vs Equally Spaced Interpolation
06:43 - Changing Chebyshev Interpolation Interval
07:34 - Chebyshev Approximation
08:00 - Chebyshev Polynomials Orthogonality
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Chebyshev
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MATLAB Chebyshev Polynomials
Chebyshev Approximation
Chebyshev Polynomials Orthogonality
#calculus #polynomials #mathematics

[@imeric7817]

Great video with nice examples!

[@LaplaceAcademy]

Glad you liked it!

[@bradlanducci9011]

Very informative, thanks!

[@LaplaceAcademy]

You're welcome! I'm glad you liked it.

[@karla1995ize]

Excelente contenido Bro

[@justinsostre8470]

The reason your recursion is slow is because you used a naive approach instead of, perhaps, tail recursion.

[@LaplaceAcademy]

But anyways, there always exists a faster way than recursion of any type.

[@holyshit922]

@@LaplaceAcademy there exists formula for Chebyshov polynomials Here is how i found it 1. From cosine of sum and cosine of difference derive recurrence relation for Chebyshov polynomials 2. Define exponential generating functiom 3. Plug it in recurrence relation 4. Solve nonhomogeneous linear differential equation of second order with constant coefficients and initial conditions (Here suitable method is Laplace transform in my opinion) 5. Calculate second derivative of solution of thiequation in step 4 6. Use Leibniz product rule to calculate nth derivative of exponential generating function of Chebyshov polynomial and evaluate it at zero After these steps we should get T_{n}(x) = sum({n \choose 2k}x^{n-2k}(x^2-1)^k,k=0..floor(n/2)) If we use binomial expansion to the (x^2-1)^k we will get T_{n}(x) = sum(binomial(n,2k)x^{n-2k}sum(binomial(k,m)x^{2m}(-1)^{k-m},m=0..k),k=0..floor(n/2)) So we will get double sum T_{n}(x) = sum(sum((-1)^{k-m}binomial(n,2k)binomial(k,m)x^{n+2m-2k},m=0..k),k=0..floor(n/2)) But how to calculate further I'm sure that it is possible to get single sum for all coefficients of Chebyshov polynomial but i dont know how to do it