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Page 872 of 165503

  • Preface

  • James Bagrow, University of Vermont, Yong‐Yeol Ahn, Indiana University, Bloomington
  • Book:
    Working with Network Data
    Published online:
    06 June 2024
    Print publication:
    13 June 2024, pp xi-xiv

  • Summary

    The fundamental practices and principles of network data are presented in this book, and the preface serves as an important starting point for readers to understand the goals and objectives of this text. The preface explains how the practical and fundamental aspects of network data are intertwined, and how they can be used to solve real-world problems. It also gives advice on how to use the book, including the boxes that will be featured throughout the book to highlight key concepts and provide practical examples of working with network data. Readers will find this preface to be a valuable resource as they begin their journey into the world of network science.

  • 17 - Mathematical Intricacies*

  • P. P. Vaidyanathan, California Institute of Technology
  • Book:
    Signals, Systems, and Signal Processing
    Published online:
    29 August 2024
    Print publication:
    13 June 2024, pp 638-670

  • Summary

    This chapter presents mathematical details relating to the Fourier transform (FT), Fourier series, and their inverses. These details were omitted in the preceding chapters in order to enable the reader to focus on the engineering side. The material reviewed in this chapter is fundamental and of lasting value, even though from the engineer’s viewpoint the importance may not manifest in day-to-day applications of Fourier representations. First the chapter discusses the discrete-time case, wherein two types of Fourier transform are distinguished, namely, $\ell_1$-FT and $\ell_2$-FT. A similar distinction between L1-FT and L2-FT for the continuous-time case is made next. When such FTs do not exist, it is still possible for a Fourier transform (or inverse) to exist in the sense of the so-called Cauchy principal value or improper Riemann integral, as explained. A detailed discussion on the pointwise convergence of the Fourier series representation is then given, wherein a number of sufficient conditions for such convergence are presented. This involves concepts such as bounded variation, one-sided derivatives, and so on. Detailed discussions of these concepts, along with several illuminating examples, are presented. The discussion is also extended to the case of the Fourier integral.

Page 872 of 165503