The first edition of Real Analysis and Foundations gave students the appropriate combination of authority, rigor, and readability that made the topic accessible while retaining the strict discourse necessary to advance their understanding. This second edition maintains this feature that made the first edition enormously popular, while further integrating new concepts built on Fourier analysis and ideas about wavelets, indicating their application to the theory of signal processing. Surpassing a purely theoretical study, the author brings relevance to the material by emphasizing the applications of real analysis in concrete engineering problems in higher dimensions.