Introduction to Chebyshev Polynomials

Chebyshev polynomials of the first kind, denoted $T_n$, occupy a prominent place in approximation theory and numerical analysis. They arise naturally in the study of trigonometric identities and minimize the maximum error when approximating functions over $[-1,1]$. This minimax property means Chebyshev polynomials oscillate between $-1$ and $1$ with the smallest possible amplitude growth for a given degree. Engineers and mathematicians leverage these polynomials when designing algorithms for interpolation, spectral methods, and digital signal processing.

Definition Using Cosine

The most direct definition uses the cosine function. For a non-negative integer $n$, we set $T_n\left(x\right)=n\text{arccos}\left(x\right)$. This relationship immediately reveals that $T_0$ equals $1$ and $T_1$ equals $x$. Because cosine is periodic, higher-degree polynomials oscillate increasingly rapidly across the interval, yet remain bounded in magnitude by one. Plotting $T_{\mathrm{n}}$ reveals $n$ zero crossings between $-1$ and $1$, a pattern that underpins their interpolation capabilities.

Recurrence Relation

Chebyshev polynomials satisfy a simple three-term recurrence that facilitates efficient computation. Specifically, $T_0=1,T_1=x,T_{n+1}=2x{T}_n-T_{n-1}$. Implementing this recurrence in code ensures numerical stability and avoids evaluating expensive trigonometric functions repeatedly. The calculator below uses precisely this relation, iterating from $0$ up to the requested degree.

Orthogonality and Weighting

A remarkable property is that these polynomials are orthogonal on $[-1,1]$ with respect to the weight $\frac{1}{\sqrt{1-x^2}}$. Formally, $\underset{-1}{\overset{1}{\int }}{T}_m\left(x\right)T_n\left(x\right)\frac{\mathrm{dx}}{\sqrt{1-x^2}}$ equals zero whenever $m$ differs from $n$. This orthogonality underpins spectral methods for differential equations because it allows functions to be represented as sums of Chebyshev modes with independent coefficients.

Minimax Behavior

Among all monic polynomials of degree $n$, the scaled Chebyshev polynomial $\frac{T_n}{2^{n-1}}$ has the smallest maximum magnitude on $[-1,1]$. This minimax property explains why they form the basis of Chebyshev approximation theory. In practice, approximating a smooth function by a sum of Chebyshev polynomials tends to minimize the largest error, creating uniform accuracy across the interval rather than focusing on one region at the expense of another.

Applications in Approximation

Chebyshev polynomials drive many state-of-the-art approximation strategies. For instance, polynomial interpolation at Chebyshev nodes dramatically reduces the Runge phenomenon—a problem where high-degree polynomials oscillate wildly near the endpoints when using equally spaced points. Because the nodes cluster more densely near the interval ends, interpolation remains stable even at high degrees. Additionally, Chebyshev expansions are often used to create spectral methods for solving differential equations. By representing the solution as a sum of Chebyshev polynomials and enforcing the equation at specific collocation points, these methods achieve exponential convergence for smooth problems.

Roots and Interpolation Points

The roots of $T_n$ are given by $x_k=\frac{2k-1}{2n}\pi$ for $k=1$ to $n$. These points are precisely the Chebyshev nodes used for interpolation and quadrature. Integrating functions by Gauss–Chebyshev quadrature involves evaluating at these roots with specific weights, offering tremendous accuracy for smooth integrands.

Algorithmic Implementation

The calculator's code follows the recurrence relation above. Starting with $T_0=1$ and $T_1=x$, the script iteratively computes higher values using the formula $T_{k+1}=2x{T}_k-T_{k-1}$. Because this approach only requires multiplication and subtraction, it is both fast and numerically stable. We also provide the alternate expression $T_n\left(x\right)=n\text{arccos}\left(x\right)$ for cross-checking results.

Worked Example

Suppose $n=3$ and $x=0.5$. We have $T_0=1$, $T_1=0.5$, and $T_2=2\left(0.5\right)T_1-T_0=-0.5$. One more step yields $T_3=2\left(0.5\right)T_2-T_1=-1.5$. This matches the cosine definition as $3\text{arccos}\left(0.5\right)=-1.5$.

Practical Tips

When using Chebyshev polynomials for approximation, scale and translate your variable if the domain differs from $[-1,1]$. Many applications remap an interval $[a,b]$ to this standard range using $x=\frac{2t-a-b}{b-a}$. Additionally, large degrees can amplify round-off errors, so high precision arithmetic might be required for extremely sensitive computations.

Further Reading

The study of Chebyshev polynomials extends into many related areas, including Chebyshev filters in signal processing, Clenshaw’s algorithm for stable evaluation, and Chebyshev–Gauss quadrature for accurate integration. Exploring these topics reveals how a seemingly simple family of polynomials connects with complex practical problems.

Historical Note

Pafnuty Chebyshev introduced these polynomials in the nineteenth century while investigating numerical approximation and probability. Their influence has grown steadily since, shaping fields as diverse as celestial mechanics and computer graphics. By experimenting with the calculator and reading more about their properties, you join a mathematical tradition that spans over a century of discovery.