orthogonal system

In ordinary space, x-, y- and z-axes are useful precisely because they are perpendicular: each measures a direction the others ignore. An orthogonal system is the same idea for functions — a family whose members are mutually ‘perpendicular’ under an integral inner product, so each one captures a feature independent of the rest. This is what makes expansions in such a system simple: you read off each coefficient without untangling it from the others.

Concretely, in an inner product space (such as L2 with inner product of f and g equal to integral of f times conjugate of g), a system (phi_n) is orthogonal if inner product of phi_n and phi_m = 0 whenever n is not equal to m. If in addition each phi_n has norm 1, the system is orthonormal. The trigonometric functions {1, cos n x, sin n x} on [-pi, pi], and the complex exponentials {e^{i n x}}, are the prototype, satisfying integral over [-pi, pi] of e^{i n x} times e^{-i m x} dx = 0 for n not equal to m.

Orthogonality alone is not enough to represent every function — you also need completeness, meaning no nonzero function is orthogonal to the entire system. A merely orthogonal system can miss directions, giving only Bessel's inequality; a complete orthonormal system spans the whole space, yielding equality (Parseval) and genuine series expansions. Other important complete orthogonal systems include Legendre, Hermite, and Laguerre polynomials, each tailored to a different weight and interval.