Question

Discuss the advantages and disadvantages between- two orthogonal Chebyshev and

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Answer 1

Explanation:

The Chebyshev polynomials are two sequences of polynomials related to the sine and cosine functions, notated as {\displaystyle T_{n}(x)}T_n(x) and {\displaystyle U_{n}(x)}U_n(x). They can be defined several ways that have the same end result; in this article the polynomials are defined by starting with trigonometric functions:

The Chebyshev polynomials of the first kind {\displaystyle T_{n}(\theta )}{\displaystyle T_{n}(\theta )} are given by

{\displaystyle T_{n}\left(\cos {\theta }\right)=\cos {(n\theta )}.}{\displaystyle T_{n}\left(\cos {\theta }\right)=\cos {(n\theta )}.}

Similarly, define the Chebyshev polynomials of the second kind {\displaystyle U_{n}(\theta )}{\displaystyle U_{n}(\theta )} as

{\displaystyle U_{n}\left(\cos {\theta }\right)\sin {\theta }=\sin {{\big (}{\big (}n+1)\theta {\big )}}.}{\displaystyle U_{n}\left(\cos {\theta }\right)\sin {\theta }=\sin {{\big (}{\big (}n+1)\theta {\big )}}.}

These definitions do not appear to be polynomials, but by using various trigonometric identities they can be converted to an explicitly polynomial form. For example, for n = 2 the T2 formula can be converted into a polynomial with argument x = cos(θ), using the double angle formula:

{\displaystyle \cos(2\theta )=2\cos ^{2}(\theta )-1.}{\displaystyle \cos(2\theta )=2\cos ^{2}(\theta )-1.}

Replacing the terms in the formula with the definitions above, we get

T2(x) = 2 x2 − 1.