Parseval's Identity

Real Analysis · Axiom Academy

The infinite-dimensional Pythagorean theorem: a function's energy equals the sum of the energies of its Fourier coefficients. 1. The Identity: Energy Is Conserved Write a function f on as its Fourier series. Parseval's Identity says the total energy on the left — the integral of |f|^2 — is exactly the energy held in the coefficients on the right. left side: the function's energy (an integral) right side: the coefficients' energy (a sum of squares) 2. Why It's True: Pythagoras, Forever The reason is geometry. On the functions are orthogonal : any two distinct ones integrate to zero against each other. They act like perpendicular axes, and f is a vector with one component along each axis. The trig functions are mutually perpendicular in L^2 — distinct ones have inner product zero. Each a_n, b_n is the component of f along one basis direction — its coordinate on that axis. Pythagoras: the squared length of f is the sum of its squared components — no cross terms survive. The only new idea is taking the theorem to infinitely many dimensions, one per frequency. Expand using the series. Orthogonality kills every cross term, leaving only the squared coefficients — that sum is Parseval's right-hand side. 3. A Famous Payoff: Choose f(x) = x Parseval becomes a calculator for infinite series once you feed it the right function. To attack the Basel problem — the value of — take the simplest odd function, f(x) = x on .

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