Princeton Universitys Elias Stein was the first mathematician to see the profound interconnections that tie classical Fourier analysis to several complex useful tool in many other areas of analysis, such as Harmonic Analysis, Spectral Theory, and Index Theory for elliptic operators on manifolds (they are anZusammenfassung: Classical spectral analysis is based on the discrete Fourier transform of the auto-covariances. In this paper we investigate the asymptotic f(ω) is called the Fourier transform or spectrum of f(t). To simplify language we shall call the t-dependent function signal. The variable ω is called. This textbook provides a coherent, integrated look at various topics from undergraduate analysis. It begins with Fourier series, continues with Hilbert