Functions, Spaces, and Expansions (Applied and Numerical Harmonic Analysis)

This graduate-level textbook presents a detailed exposition of key mathematical tools in analysis, which will appeal to students and professionals across science and engineering. Every topic covered has been specifically chosen because it plays a role outside the field of pure mathematics, so although the treatment of each is mathematical in nature, and concrete applications are not delineated, the principles and tools presented are quite useful when exploring the computational areas of physics and engineering. A central theme of the textbook is the structure of various vector spaces (most importantly, Hilbert spaces) and expansions of elements in these spaces in terms of bases. Particular attention is given to the space of square-integrable functions, L2[registered]. Key topics and features include: more than 150 exercises; an introduction to the idea of the mathematical proof; abstract vector spaces; normed vector spaces; approximation in normed vector spaces; Banach spaces; Hilbert spaces, including linear operators and orthonormal bases; the Fourier transform, including the discrete Fourier transform; wavelet and multiresolution analysis; B-Splines; Sturm-Liouville problems; and, a comprehensive bibliography. As a textbook that provides a deep understanding of central issues in mathematical analysis, “Functions, Spaces, and Expansions” is intended for graduate students, researchers, and practitioners in applied mathematics, physics, and engineering. Readers are expected to have a solid understanding of linear algebra, in Rn and in general vector spaces. They should also be familiar with the basic concepts of calculus and real analysis, including Riemann integrals and the infinite series of real or complex numbers.