Lecture 7: Fourier Series

This lecture introduces Fourier series as a tool for representing periodic functions using sums of sines and cosines. The orthogonality of trigonometric functions is established, leading to formulas for the Fourier coefficients. Dirichlet's theorem provides convergence conditions: at continuity points the series converges to the function value, and at jump discontinuities it converges to the average of the left and right limits. Worked examples include square wave and sawtooth wave expansions, along with even and odd extensions to obtain cosine or sine series. Parseval's identity connects the squared integral of a function to the sum of squares of its Fourier coefficients, illustrated by evaluating the sum of reciprocals of odd squares. The lecture concludes with applications in signal processing and differential equations.