Chebyshev polynomials T_n(x) = cos(n·arccos(x)) on [−1,1] satisfy the equioscillation theorem: the minimax polynomial of degree n equioscillates at n+2 points.
Chebyshev nodes: x_k = cos((2k+1)π/(2n)) nearly optimal interpolation nodes — suppress Runge's phenomenon by clustering at endpoints.
Monomial interpolation on equidistant nodes is ill-conditioned: Lebesgue constant grows exponentially → Runge's phenomenon for smooth functions on large intervals.