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Harmonic Analysis and Applications

Michael Th. Rassias, ed.
Publication Date: 
Number of Pages: 
Springer Optimization and its Applications
[Reviewed by
Brody Johnson
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The objective of this contributed volume is to summarize the state-of-the-art in a wide variety of research areas where harmonic analysis finds application. As such, the primary audience for this volume will be researchers who work at the interface of harmonic analysis and one or more of the represented application areas, although some effort has been made to make the content accessible to graduate students. The volume is unlikely to be used as a textbook; however, it might serve as the foundation for the occasional graduate topics course in harmonic analysis. Many of the contributed chapters present new results and most provide a thoughtful introduction along with sufficient references to orient readers unfamiliar with the relevant subfield.
What follows is a brief description of each contributed chapter, in order of appearance.
  • “Sampling and Approximation in Shift Invariant Subspaces of \( L_{2}(\mathbb{R}) \)” by Nikolaos Atreas.  This chapter develops uniform and non-uniform sampling expansions (along with other related results) for shift-invariant spaces \( V_{\phi} \subset L_{2}(\mathbb{R}) \) under certain admissibility conditions on the generator \( \theta \). The author makes use of Wiener’s lemma for operator algebras as well as the finite section method.
  • “Optimal ` 1 Rank One Matrix Decomposition” by Radu Balan, Kasso A. Okoudjou, Michael Rawson, Yang Wang, and Rui Zhang.  The authors introduce various measures of optimality for the decomposition of positive semidefinite matrices as sums of rank one matrices. Optimal decompositions are presented for diagonally dominant matrices, 2 × 2 matrices, and a class of 3 × 3 matrices.
  • “An Arithmetical Function Related to Báez-Duarte’s Criterion for the Riemann Hypothesis” by Michael Balazard.  This primarily expository chapter reexamines the work of Báez-Duarte, starting from Weingartner’s formulation and introducing an arithmetical function ν that equals the Möbius function if and only if the Riemann Hypothesis holds.
  • “Large Sets Avoiding Rough Patterns” by Jacob Denson, Malabika Pramanik, and Joshua Zahl.  The authors offer new, constructive results for rough pattern avoidance in geometric measure theory. The pattern avoidance problem seeks to understand the existence and dimension of subsets of \( \mathbb{R}^{d} \) that avoid a specified pattern, e.g., a non-trivial arithmetic progression.
  • “PDE Methods in Random Matrix Theory” by Brian Hall.  The ultimate goal of this chapter is to describe the notion of a Brown measure and demonstrate the computation of Brown measures using methods from partial differential equations. The chapter includes preliminary sections on random matrices and the connection between random matrix models and operator algebras.
  • “Structure and Optimisation in Computational Harmonic Analysis: On Key Aspects in Sparse Regularisation” by Anders C. Hansen and Bogdan Roman.  This chapter examines various tests aimed at demonstrating a relationship between the success of sparse recovery algorithms and key structures in the signals, going beyond simple sparsity.  This includes a generalized flip test to investigate the sensitivity of sparse recovery algorithms to changes in the structure of the sparse signal.
  • “Reflections on a Theorem of Boas and Pollard” by Christopher Heil.  The author presents an expository chapter examining the multiplicative completion of redundant systems in Hilbert and Banach function spaces. This includes an extension of the Boas and Pollard result as well as a brief account of related results found in the literature.
  • “The Andoni-Krauthgamer-Razenshteyn Characterization of Sketchable Norms Fails for Sketchable Metrics” by Subhash Khot and Assaf Naor.  This chapter settles a question about the Andoni-Krauthgamer-Razenshteyn characterization of sketchable norms, showing that the characterization can fail when the underlying normed space is replaced by a metric space.
  • “Degree of Convergence of Some Operators Associated with Hardy-Littlewood Series for Functions of Class \( Lip(\alpha,p), p > 1 \)” by Manish Kumar, Benjamin A. Landon, R.N. Mohapatra, and Tusharakanta Pradhan.  The authors examine the degree of convergence of the Hardy-Littlewood series of a function \( f \) to itself in \( H_{\alpha,p} \) for three specific means.
  • “Real Variable Methods in Harmonic Analysis and Navier-Stokes Equations” by Pierre Gilles Lemariè-Rieusset.  The author demonstrates the usefulness of real variable methods in harmonic analysis for the study of nonlinear partial differential equations through a discussion of several modern approaches to the Navier-Stokes equations.
  • “Explore Intrinsic Geometry of Sleep Dynamics and Predict Sleep Stage by Unsupervised Learning Techniques” by Gi-Ren Liu, Yu-Lun Lu, Yuan-Chung Shen, and Hau-Tieng Wu.  In this chapter, the authors propose an unsupervised learning approach for the understanding of sleep dynamics and the identification of sleep stage. This approach makes use of specific diffusion-based algorithms that act on spectral information taken from the synchrosqueezing transform of an electroencephalogram.
  • “Harmonic Functions in Slabs and Half-Spaces” by W.R. Madych.  The author examines harmonic functions on the slab \( \mathbb{R}^{n} \times (a,b) (-\infty < a < b < \infty) \) and the half-space \( \mathbb{R}^{n} \times (0, \infty) \). This work includes a characterization of harmonic functions on the slab which exhibit no greater than polynomial growth.

Brody Johnson is an Associate Professor in the Department of Mathematics and Statistics at Saint Louis University in St. Louis, Missouri, USA.