This book provides a treatment of analytical methods of applied mathematics. It starts with a review of the basics of vector spaces and brings the reader to an advanced discussion of applied mathematics, including the latest applications to systems and control theory. The text is designed to be accessible to those not familiar with the material and useful to working scientists, engineers, and mathematics students. The author provides the motivations of definitions and the ideas underlying proofs but does not sacrifice mathematical rigor.

It presents:

• an easily accessible discussion of analytical methods of applied mathematics from vector spaces to distributions, Fourier analysis, and Hardy spaces with applications to system theory;
• an introduction to modern functional analytic methods to better familiarize readers with basic methods and mathematical thinking; and
• an understandable yet penetrating treatment of such modern methods and topics as function spaces and distributions, Fourier and Laplace analyses, and Hardy spaces.

Keywords: vector spaces, distributions, Fourier analysis, and Hardy spaces, functional analytic methods, systems and control theory

Yutaka Yamamoto is a professor in the Department of Applied Analysis and Complex Dynamical Systems, Graduate School of Informatics, Kyoto University, Kyoto, Japan. His main interest is in system and control theory, particularly the theory of sampled data control systems, its application to digital signal processing, realization and robust control of distributed parameter systems, and related mathematical problems. He is a recipient of the G. S. Axelby Outstanding Paper Award of the IEEE Control Systems Society, the Commendation for Science and Technology by the Minister of Education in Japan, the Distinguished Member Award of the Control Systems Society of the IEEE, and various other awards. He has authored or co-authored more than 200 journal or conference papers. He is currently President-Elect of the IEEE Control Systems Society and past President of ISCIE Japan. He is a fellow of the IEEE and ISCIE, Japan.

Preface; Glossary of Notation;  1 Vector Spaces Revisited; 2 Normed Linear Spaces and Banach Spaces; 3 Inner Product and Hilbert Spaces; 4 Dual ; 5 The Space L(X,Y) of Linear Operators; 6 Schwartz Distributions; 7 Fourier Series and Fourier Transform; 8 Laplace Transform; 9 Hardy Spaces; 10 Applications to Systems and Control; Appendix A Some Background on Sets, Mappings, and Topology; Appendix B Table of Laplace Transforms; C Solutions; D Bibliographical Notes; Bibliography; Index