Скачать или смотреть Lecture 4: Chebyshev Polynomials (explained in details)

In this video entitled Lecture 4: Chebyshev Polynomials (explained in details), we define and explain in details what Chebyshev polynomials are all about.

We set up the stage by first introducing the concept of generalized least squares, that is, the approximating function is written as a linear combination of the elements of a basis of a space of functions. We then show that if the basis is orthogonal or orthonormal, the least squares problem is simplified, that is, there is no need to solve the dense system of linear equations provided by the Normal Equations (by Gauss Elimination or related methods), since the solutions are readily obtained.

Once the stage has been set, the natural choice of orthogonal polynomials is Chebyshev polynomials. We define and give an algebraic expression of Chebyshev polynomials, then use trigonometric identities to generate a recursion formula for Chebyshev polynomials of the first kind. After computing some few polynomials, we show that a Chebyshev polynomial of degree n has exactly n roots.

Finally, we give the orthogonality properties of Chebyshev polynomials and state a related theorem that gives the expression of the coefficients of the Chebyshev approximation polynomial (i.e. the least squares solution).

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Lecture 3: Continued Fractions for Rational Padé Approximations
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Chapters/Timestamps
00:04 Introduction to the generalized least squares method
07:08 Orthonormality properties
13:08 Simplification of the normal equations
16:07 Introduction to Chebyshev polynomials
19:25 Recursion formula and its derivation via trigonometry identities
28:47 Algebraic expressions of few Chebyshev polynomials
30:00 Roots of Chebyshev polynomials
53:00 Chebyshev approximation polynomial
54:46 Orthogonality property of Chebyshev polynomials
01:00:30 Chebyshev approximation theorem and application