Chebyshev Polynomials 

In numerical analysis and approximation theory, one of the fundamental problems is to approximate a continuous function by a polynomial. Polynomial interpolation allows us to construct a polynomial \(p_n(x)\) of degree \(n\) that passes exactly through a given set of points \((x_i, f(x_i))\). While this is a straightforward concept, the choice of interpolation points is critical for the quality and stability of the approximation. Poorly chosen points can lead to large oscillations of the polynomial near the ends of the interval, a phenomenon known as Runge's phenomenon.

To address this, mathematicians have introduced a special class of points called Chebyshev points, which are the roots of Chebyshev polynomials of the first kind. These points are distributed more densely near the endpoints of the interval, helping to reduce oscillations and improve the accuracy of the interpolation. Chebyshev points are not only theoretically optimal—they also provide excellent practical performance in applications such as numerical integration, spectral methods for differential equations, and function approximation.

Chebyshev polynomials themselves, denoted \(T_n(x)\), form a family of polynomials with unique properties: they oscillate between \(-1\) and \(1\) on \([-1,1]\), have well-defined extrema and roots, and satisfy a simple recurrence relation. Their remarkable properties make them the ideal tool for constructing interpolation nodes that minimize the maximum error across the interval. In other words, when using Chebyshev points, the interpolating polynomial achieves the smallest possible "worst-case" deviation from the original function among all choices of \(n+1\) points.

This introduction will explore both the **Chebyshev polynomials** and their use in **polynomial interpolation**, providing formal definitions, properties, and practical examples. By the end of this section, students will understand why Chebyshev polynomials are central to numerical approximation, how to compute Chebyshev points, and how these points optimize interpolation accuracy compared to equally spaced points.

Furthermore, we will illustrate these concepts with visual examples showing how interpolation using Chebyshev points avoids oscillations near the boundaries, demonstrating both the theory and its practical benefits.