Spectral Properties of Fourier Series and Fourier Transforms

This lecture covers the motivation behind Fourier series and Fourier transforms, emphasizing their applications in solving partial differential equations. It delves into the fundamentals of complex exponential Fourier series, energy spectral density, even and odd periodic triangular waves, translation properties, and the Fourier transform for non-periodic signals. The instructor discusses the universal spectra, the definition of the Fourier transform, and its applications for non-periodic functions, including solving unbounded ODEs and PDEs conveniently in Fourier transform space.