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2 results

  • 8 - Applications

  • from Part III - Discrete Functional Analysis
  • Martin Buntinas, Loyola University, Chicago
  • Book:
    Classical and Discrete Functional Analysis with Measure Theory
    Published online:
    06 January 2022
    Print publication:
    20 January 2022, pp 333-402

  • Summary

    Many applications of Functional Analysis are introduced, including Least Squares Approximation Methods, the Vibrating String or Membrane (the Wave Equation), Heat Flow on a rod or plate (the Heat Equation), Gambler's Ruin and Random Walk, Sampling Theorem of Signal Processing, the Atomic Theory of Matter, Uncertainty Principle, and Wavelets. The beautiful connection between Group Theory, Fourier Series, and the Haar Integral (which for Euclidean Space, is the Lebesgue Integral) is investigated.

Classical and Discrete Functional Analysis with Measure Theory
  • Martin Buntinas
  • Published online:
    06 January 2022
    Print publication:
    20 January 2022
  • Functional analysis deals with infinite-dimensional spaces. Its results are among the greatest achievements of modern mathematics and it has wide-reaching applications to probability theory, statistics, economics, classical and quantum physics, chemistry, engineering, and pure mathematics. This book deals with measure theory and discrete aspects of functional analysis, including Fourier series, sequence spaces, matrix maps, and summability. Based on the author's extensive teaching experience, the text is accessible to advanced undergraduate and first-year graduate students. It can be used as a basis for a one-term course or for a one-year sequence, and is suitable for self-study for readers with an undergraduate-level understanding of real analysis and linear algebra. More than 750 exercises are included to help the reader test their understanding. Key background material is summarized in the Preliminaries.