Advanced Engineering Mathematics, Lecture 3.8: Pythagoras, Parseval, and Plancherel

Advanced Engineering Mathematics, Lecture 3.8: Pythagoras, Parseval, and Plancherel.

The ancient Greek theorem of Pythaorgas says that for any vector v in R^n, the square of its norm is just the sum of the squares of the coefficients (with respect to the standard basis). This also remarkably holds for periodic functions: ||f||^2 is the infinite sum of the c_n's, its complex Fourier coefficients. That result is known as Parseval's identity, and the "continuum" analogue of it is known as Plancherel's theorem, which says that a function and its Fourier transform have the same norm. In fact, more is true: the Fourier transform also preserves inner products. We prove these statements, and then conclude with a neat application of using the real Fourier version of Parseval's identity and an odd sawtooth wave to compute the inverse squares of the positive integers.