The Chebyshev polynomials: (a) Prove that Tn(t)=cos(n arccost), n=0,1,2, …, form a system of ort

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The Chebyshev polynomials: (a) Prove that Tn(t)=cos(n...

The Chebyshev polynomials: (a) Prove that $T_n(t)=\cos (n \arccos t), n=0,1,2, \ldots$, form a system of orthogonal polynomials under the weighted inner product $$ \langle f, g\rangle=\int_{-1}^1 \frac{f(t) g(t) d t}{\sqrt{1-t^2}} . $$ (b) What is $\left\|T_n\right\|$ ? (c) Write out the formulas for $T_0(t), \ldots, T_6(t)$ and plot their graphs.

The Chebyshev polynomials: (a) Prove that $T_n(t)=\cos (n \arccos t), n=0,1,2, \ldots$, form a system of orthogonal polynomials under the weighted inner product
$$
\langle f, g\rangle=\int_{-1}^1 \frac{f(t) g(t) d t}{\sqrt{1-t^2}} .
$$
(b) What is $\left\|T_n\right\|$ ? (c) Write out the formulas for $T_0(t), \ldots, T_6(t)$ and plot their graphs.

Key Concepts

Trigonometric Representation and Cosine Formula

The use of the trigonometric representation T?(t) = cos(n arccos t) not only defines the Chebyshev polynomials but also provides insight into their oscillatory behavior and structural properties. This representation makes it possible to prove their orthogonality through the properties of the cosine function and the integration over a transformed variable, revealing deeper connections between polynomial approximation and Fourier analysis.

Weighted Inner Product

A weighted inner product on a function space includes an additional weight function in the integration process, modifying the inner product by giving different importance, or weight, to different parts of the domain. In the context of Chebyshev polynomials, the weight function (1/?(1-t²)) is used, leading to the specific orthogonality properties that are central to their applications in numerical methods.

Norm in Function Spaces

The norm of a polynomial in a function space endowed with an inner product is a measure of its size and is typically defined as the square root of the inner product of the polynomial with itself. Computing the norm of Chebyshev polynomials with respect to the weighted inner product helps in normalizing these functions for various applications, such as constructing orthonormal bases for function approximation.

Chebyshev Polynomials

Chebyshev polynomials are a classical sequence of polynomials defined by a trigonometric formula, T?(t) = cos(n arccos t), which links them to the cosine function. They play a significant role in approximation theory, spectral methods, and numerical analysis because of their extremal properties and minimized maximum error in polynomial interpolation.

Orthogonal Polynomials

Orthogonal polynomials are a system of polynomials in which any two different polynomials are orthogonal with respect to a given inner product, meaning their inner product evaluates to zero. This property is extremely useful in function approximation and solving differential equations via spectral methods, as it allows decomposing functions into components that can be separately analyzed.

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