References & Further Reading

References

  1. E. Kreyszig, Introductory Functional Analysis with Applications, 1978

    The classic engineering-friendly text. Accessible treatment of normed spaces, Banach and Hilbert spaces, and operator theory with applications throughout.

  2. H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, 2011

    Modern treatment bridging pure and applied. Excellent coverage of Sobolev spaces and distributions relevant to Sections 1.5 and 1.6.

  3. H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, 1996

    The standard reference for ill-posed problems. Chapters 2\u20133 cover the spectral theory and Picard condition we develop in Sections 1.4 and 1.6.

  4. A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, 2011

    Accessible introduction connecting functional analysis to inverse problems. Bridges the gap between our Chapter 1 foundations and Chapter 2 regularization.

  5. J. B. Conway, A Course in Functional Analysis, 1990

    Complete treatment of operator theory. Chapters on compact operators and spectral theory go well beyond our coverage for readers wanting full proofs.

  6. N. Young, An Introduction to Hilbert Space, 1988

    Concise and focused on Hilbert space theory. Ideal companion for Sections 1.2\u20131.3 on inner product spaces and bounded operators.

  7. G. Caire, Illumination-Sensing for RF Imaging: A Unified Model, 2023

    Establishes the unified forward model adopted throughout this book. Derives the sensing matrix from first principles using the Born approximation and shows how system geometry (antenna placement, waveform, carrier frequency) determines the singular value structure of the sensing operator.

Further Reading

For readers who want to go deeper into specific topics from this chapter.

  • Operator theory for signal processing

    S. Mallat, *A Wavelet Tour of Signal Processing*, 3rd ed., Academic Press, 2009

    Multiresolution analysis bridges functional analysis and practical imaging. Wavelet bases provide the $L^2$ decompositions from Section 1.2 in a form optimized for localized features in images.

  • Inverse problems survey

    S. Arridge et al., \u2018Solving inverse problems using data-driven models,\u2019 Acta Numerica, vol. 28, pp. 1\u2013174, 2019

    Modern perspective connecting the classical ill-posedness theory of Section 1.6 to deep learning approaches. Essential context for Chapters 8\u201310.

  • Imaging mathematics

    O. Scherzer et al., *Variational Methods in Imaging*, Springer, 2009

    Comprehensive treatment of functional analysis applied specifically to imaging. Covers the forward operator framework of Section 1.6 in much greater depth.

  • Applied functional analysis

    J.-P. Aubin, *Applied Functional Analysis*, 2nd ed., Wiley, 2000

    Focuses on PDE and optimization applications relevant to Chapters 2\u20134. Provides the Sobolev space theory underlying regularization in function spaces.

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