Chebyshev Polynomials

For -1 ≤ x ≤ 1, T_n(x) = cos(n arccos x). For n ≥ 1 the n roots are cos((2k-1)π/(2n)), k=1,…,n, and the n+1 alternating extrema occur at cos(kπ/n), k=0,…,n. The constant T₀=1 has no roots. Endpoint derivatives come from the polynomial recurrence, so no division by √(1-x²) is needed.

Worked Example: The Cubic Case

Let's find the explicit form of the Chebyshev polynomial of degree 3, T₃(x).

Using the recurrence relation T_n+1(x) = 2xT_n(x) - T_n-1(x) with T₀(x) = 1 and T₁(x) = x:

T₂(x) = 2x(x) - 1 = 2x² - 1
T₃(x) = 2x(2x² - 1) - x = 4x³ - 3x

The leading coefficient is 4, which is 2^3-1. The monic polynomial is obtained by dividing by 4, giving x³ - (3/4)x. The supremum norm of this monic polynomial on [-1, 1] is 1/4.

Interactive Experiment

You can interactively change the degree of the Chebyshev polynomial and the probe point in the module.

Set the Degree n control to 3.
Set the Probe x control to 0.5.

You will observe that the polynomial evaluation result T₃(x) gives -1, which matches the calculation T₃(0.5) = 4(0.5)³ - 3(0.5) = 4(0.125) - 1.5 = 0.5 - 1.5 = -1. Also note that at this point the derivative result shows 0, indicating an extremum.

Try exploring other combinations in the interactive experiment.

Limitations

This visualization only plots Chebyshev polynomials of the first kind up to degree 8. Higher degrees have more densely spaced roots (Chebyshev nodes) and extrema, which become harder to distinguish on a small screen. Additionally, this module restricts the domain to the standard interval [-1, 1]; it does not demonstrate the exponential growth of the polynomials outside this interval.

References

For more detailed mathematical properties, refer to the following sources from the NIST Digital Library of Mathematical Functions:

Chebyshev Polynomials