Uniqueness of Fourier Representations

This lecture delves into the uniqueness of Fourier representations, exploring the extension of Fourier series to continuous functions on the circle. The instructor discusses the concept of approximate identity, introducing the Cesàro means and the Schwartz derivative. Through a detailed proof, the lecture demonstrates that if a function is represented by two different Fourier series, both valid at every point, then the coefficients must be equal. The proof involves controlling the coefficients, showing that they converge to zero. The lecture also touches upon the convergence properties of Fourier series for continuous functions, highlighting the importance of the uniqueness theorem in harmonic analysis. The instructor provides insights into the historical context of the theorem, its implications, and related open questions in the field.