ABSTRACT
In this chapter, the authors review relevant concepts and results from linear algebra before they apply them to solve the least-squares problem in a more general setting. Orthogonal projections are of fundamental importance in the theory of Fourier series and in many other branches of mathematics. The authors aim to study the convergence of Fourier series as well as the Gibbs phenomenon. The faster the Fourier series coefficients converge to zero, the less impact the aliased frequencies have on the reconstructed signal. The authors seek to prove that the Fourier series of a periodic piecewise smooth function converges to the normalized function value at every point. They deal with the denition which allows to address the orthogonality of both real-valued and complex-valued functions.
