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Jacob Cox
Jacob Cox

Grafakos L. Complex Analysis With Applications

The additional inner product structure allows for a richer theory, with applications to, for instance, Fourier series and quantum mechanics. Functions in L 2 \displaystyle L^2 are sometimes called square-integrable functions, quadratically integrable functions or square-summable functions, but sometimes these terms are reserved for functions that are square-integrable in some other sense, such as in the sense of a Riemann integral (Titchmarsh 1976).

Grafakos L. Complex analysis with applications


However, I understand that as one gets deeper into a subject such as harmonic analysis, one would need to understand several related areas in greater depth such as functional analysis, PDE's and several complex variables. Therefore, suggestions of how one can incorporate these subjects into one's learning of harmonic analysis are welcome. (Of course, since this is mainly a request for a roadmap in harmonic analysis, it might be better to keep any recommendations of references in these subjects at least a little related to harmonic analysis.)

It depends very much on what areas of harmonic analysis you're interested in, of course. Grafakos' books are excellent and really quite advanced, and if you wish to continue in that style of harmonic analysis, then there's not much else you can do other than start reading many of the articles that he cites. On the other hand, there are interesting areas in harmonic analysis not covered by Grafakos. I'd recommend a couple of textbooks by Stein: Singular Integrals and Differentiability Properties of Functions and Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. There are probably some other interesting textbooks on singular integral operators that might be useful (though I can't think of any off the top of my head). One other interesting (and very modern) area is wavelets: Mayer's book Wavelets and Operators is probably the place to start there.Other useful resources are lecture notes or survey articles about harmonic analysis available online. For example, Pascal Auscher taught a course at ANU on harmonic analysis using real-variable methods last year, and one of the students in the class typed up notes, which are available here. Similarly, Terry Tao taught a course a few years ago, and he has lecture notes here and here. Finally, if you want to learn about harmonic analysis with an operator-theoretic bent, there are useful lecture notes here and here.

For connections with unitary representations: the 2nd half of Dixmier's "C*-algebras", or better R. Howe and E.C. Tan "Non-abelian harmonic analysis (applications of $SL_2(\mathbbR)$)" (everything is in the subtitle!)

One of my favourite books for harmonic analysis is Fourier analysis by Javier Duoandikoetxea. In fact, I would recommend that as your first port of call for learning harmonic analysis if you already have some background (such as an undergraduate/postgraduate course in harmonic analysis). That is a terrific reference for background regardless of what you want to do with harmonic analysis. I would tackle this before moving onto Elias Stein's book "harmonic analysis: real-variable methods, orthogonality and oscillatory integrals" (also a great book).

As mentioned above, it really depends on what type of harmonic analysis you are interested in, but I would certainly recommend those as well as harmonic analysis by Katznelson, the two volume books by Grafakos, both of Stein's books on "introduction to Fourier analysis on Euclidean spaces" and "singular integrals and differentiability properties of functions" are useful for singular integrals. I'd also recommend a treatise on trigonometric series by Bary. Zygmund's two volume books on trigonometric series are good, but I would tackle a few other books on harmonic analysis before going for it. It is quite complex in comparison to the other references and will not help much if you do not already have a foundation in harmonic/Fourier analysis.

In a more recent approach, pseudo-differential operators on (discrete pseudo-differential operators) were introduced by Molahajloo in [36], and some of its properties were developed in the last years in the references [9], [17], [28], [38], [39], [40], [41], [43]. However, only the fundamental work L. Botchway, G. Kibiti, and M. Ruzhansky [5] includes properties about a discrete pseudo-differential calculus and applications to difference equations. The reference [9] discusses those relations of the theory of discrete pseudo-differential operators with important problems in number theory as the Waring problem and the hypothesis K * by Hooley.

This textbook is intended for a one semester course in complex analysis for upper level undergraduates in mathematics. Applications, primary motivations for this text, are presented hand-in-hand with theory enabling this text to serve well in courses for students in engineering or applied sciences. The overall aim in designing this text is to accommodate students of different mathematical backgrounds and to achieve a balance between presentations of rigorous mathematical proofs and applications.

  • Besov spaces, symbolic calculus and boundedness of bilinear pseudodifferential operators, with J. Herbert. Harmonic analysis, partial differential equations, complex analysis, Banach spaces, and operator theory. Vol. 1, 275-305, Assoc. Women Math. Ser., 4, Springer, 2016.



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