ChebyshevPolynomials

The Arcsin Function and Chebyshev Polynomials of the First Kind: Exploring the Spectral Link

1. Introduction

Chebyshev polynomials of the first kind play a foundational role in mathematical analysis, with their properties closely intertwined with the arcsin function.

2. Breaking Down the Definitions

Chebyshev polynomials of the first kind, $T_n(x)$, are expressed as:

$$T_n(x) = \cos(n \cdot \arccos(x))$$

for $|x| \leq 1$.

3. Delving into Orthogonality

These polynomials exhibit orthogonality on the interval $[-1,1]$ relative to the weight function:

$$w(x) = \frac{1}{\sqrt{1-x^2}}$$

Intriguingly, when you integrate this weight function, you encounter the arcsin function:

$$\int w(x) \, dx = \arcsin(x) + C$$

where $C$ is an integration constant.

4. The Spectral Connection

The weight function, $w(x)$, serves as the spectral density, illustrating how the spectral content associated with Chebyshev polynomials spreads over $[-1,1]$. On the other hand, its integral, $\arcsin(x)$, acts as the spectral distribution function, offering a cumulative perspective of this content.

5. Why Does This Matter?

Understanding these properties offers clearer insights into approximation theory. It aids in effectively representing functions using an orthogonal basis in a specific weighted $L^2$ space, bridging the gap between abstract mathematics and practical computations.