orthogonal polynomials
Just as the x, y, z axes are perpendicular and so let you describe any point with independent, non-interfering coordinates, you can build a set of polynomials that are 'perpendicular' to each other in a function sense. Orthogonal polynomials are such a family — a tailor-made coordinate system for the space of functions, ideally suited to approximation.
Two functions are called orthogonal if a certain integral of their product (with a chosen weight) is zero — the function-space version of two vectors having zero dot product. Starting from 1, x, x^2, ... and applying Gram-Schmidt with an inner product like the integral of f(x) g(x) w(x) dx gives a sequence of polynomials p_0, p_1, p_2, ... of increasing degree that are mutually orthogonal. Different weights and intervals give the classic families: LEGENDRE polynomials (weight 1 on [-1, 1]) and CHEBYSHEV polynomials (weight 1 / sqrt(1 - x^2) on [-1, 1], and equal to cos(n arccos x)). Their power for approximation comes from orthogonality: to best-fit a function in the least-squares sense you just project onto each polynomial independently — each coefficient is computed by its own integral and does not change when you add more terms, so you can refine an approximation by simply tacking on the next term. They also satisfy a cheap three-term recurrence, making them numerically stable to generate and evaluate, unlike the treacherous raw powers x^n.
Orthogonal polynomials are a unifying thread across numerical analysis. Chebyshev polynomials connect straight back to minimax approximation (their roots are the Chebyshev nodes that tame Runge's phenomenon, and the Chebyshev series is near-best in the uniform sense); Legendre polynomials generate the nodes and weights of Gaussian quadrature, the most accurate integration rule for a given number of points; and orthogonal expansions are the backbone of spectral methods for PDEs. The honest framing: the magic is not any single polynomial but the orthogonality structure, which turns a coupled best-fit problem into a sequence of independent, stable projections — and which family to use depends on the weight that matches your problem.
The first Legendre polynomials on [-1, 1] are P_0 = 1, P_1 = x, P_2 = (3x^2 - 1)/2, P_3 = (5x^3 - 3x)/2. Check orthogonality of P_1 and P_2: the integral of x * (3x^2 - 1)/2 over [-1, 1] is 0 because the integrand is odd. Each pair integrates to zero — they form an orthogonal basis.
Mutually 'perpendicular' polynomials: independent, stable coordinates for functions.
There is no single 'orthogonal polynomial' — orthogonality is always WITH RESPECT TO a weight and interval, and each choice (Legendre, Chebyshev, Hermite, Laguerre) suits a different problem. Use the three-term recurrence to generate them, never the raw powers x^n, which are ill-conditioned.