Chebyshev Polynomials
Connect Chebyshev recurrence, roots and equal-height extrema with the least maximum size of a monic polynomial.
Free theory articleConnect Chebyshev recurrence, roots and equal-height extrema with the least maximum size of a monic polynomial.
Concept & interpretation
For -1 ≤ x ≤ 1, Tn(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.
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.