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Sorry for such a newbie post and for asking two unrelated references in one shot.

First, I am interested in any proof of Pinsker's inequality.

Second, I wonder what is the best place to read about Pontryagin duality and harmonic analysis. To clarify, I took only standard functional analysis course.

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Why not split into two questions? For the second, I recommend "Fourier Analysis on Number Fields." –  Cam McLeman Oct 18 '10 at 16:57
    
What is the point in splitting? –  ilyaraz Oct 18 '10 at 17:02
    
@ilyaraz Factorizing polynomials. –  drvitek Oct 18 '10 at 17:47
    
You might also enjoy the following paper on Generalized Pinsker Inequalities: mark.reid.name/files/pubs/colt09.pdf –  Suvrit Oct 18 '10 at 20:01
    
Cam: ilyaraz may be more of an analyst than a number theorist, so the book you recommended might not be to ilyaraz's taste. Perhaps Folland's "A Course in Abstract Harmonic Analysis" as an alternative? –  KConrad Oct 19 '10 at 5:23
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I found Rudin's "Fourier Analysis on Groups" a good reference for Pontryagin duality. The level of functional analysis there is not too high.

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Folland's book ("A Course in Abstract Harmonic Analysis") is highly extensive, developing the machinery of spectral theory, Banach algebras, topological groups and the unitary representation theory of arbitrary LCH groups before arriving at the more specific theory of (locally compact Hausdorff) abelian groups. Later he also gives very general treatments of the theory of compact groups and of the construction of induced representations, and finally, he lays down the general results known for arbitrary LCSC (locally compact, second countable and Hausdorff) groups in harmonic analysis, mostly without proof. In my opinion this book is great but it is probably too heavy for you if you're just interested in the abelian theory. In that case Katznelson's "An Introduction to Harmonic Analysis" is a nice book which may be more suitable for you.

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