This text serves as an introduction to the modern theory of analysis & differential equations with applications in mathematical physics & engineering sciences Having outgrown from a series of half-semester courses given at University of Oulu this book consists of four self-contained parts The first part Fourier Series & the Discrete Fourier Transform is devoted to the classical one-dimensional trigonometric Fourier series with some applications to PDEs & signal processing The second part Fourier Transform & Distributions is concerned with distribution theory of L Schwartz & its applications to the Schroedinger & magnetic Schroedinger operations The third part Operator Theory & Integral Equations is devoted mostly to the self-adjoint but unbounded operators in Hilbert spaces & their applications to integral equations in such spaces The fourth & final part Introduction to Partial Differential Equations serves as an introduction to modern methods for classical theory of partial differential equations Complete with nearly 250 exercises throughout this text is intended for graduate level students & researchers in the mathematical sciences & engineering