Explorations in Harmonic Analysis: With Applications to Complex Function Theory and the Heisenberg Group [Hardcover]

Harmonic analysis is a venerable part of modern mathematics. Its roots began, perhaps, with late eighteenth-century discussions of the wave equation. Using the method of separation of variables, it was realized that the equation could be solved with a data function of the form?(x)= sin jx for j? Z.Itwasnaturaltoask, using the philosophy of superposition, whether the equation could then be solved with data on the interval [0,?] consisting of a nite linear combinationof the sin jx. With an af rmative answer to that question, one is led to ask about in?nite linear combinations. This was an interesting venue in which physical reasoning interacted with mathematical reasoning. Physical intuition certainly suggests that any continuous function? can be a data function for the wave equation. So one is led to ask whether any continuous? can be expressed as an (in nite) superposition of sine functions. Thus was born the fundamental question of Fourier series. No less an eminence gris than Leonhard Euler argued against the proposition.Ontology and History of Real Analysis.- The Central Idea: The Hilbert Transform.- Essentials of the Fourier Transform.- Fractional and Singular Integrals.- A Crash Course in Several Complex Variables.- Pseudoconvexity and Domains of Holomorphy.- Canonical Complex Integral Operators.- Hardy Spaces Old and New.- to the Heisenberg Group.- Analysis on the Heisenberg Group.- A Coda on Domains of Finite Type.

From the reviews:

This is an ambitious introduction to a particular direction in modern harmonic analysis. It presents harmonic analysis in vitro  in a context in which it is actually applied complex variables and partial differential equations. This will make the learning experience more meaningful for graduate students who are just beginning to forge a path of research. (Vasily A. Chernecky, Zentralblatt MATH, Vol. 1171, 2009)

Writing a 350 page account on harmonic analysis & is no easy task and maklc¡