Why uniform nodes fail at high degree — and how Chebyshev nodes fix it
Runge's phenomenon: interpolating f(x)=1/(1+25x²) with equally-spaced points causes wild oscillations near the edges as degree n grows. Chebyshev nodes xₖ = cos((2k−1)π/2n) concentrate near ±1, minimizing the Lebesgue constant and suppressing oscillations. The Chebyshev polynomial Tₙ(x) = cos(n·arccos x) achieves the minimax property: smallest maximum error among all degree-n polynomials.