I am reading a book on Harmonic Analysis on $\mathbb R^n$. It needs some facts about finite measure space $\mathcal B(\mathbb R^n)$, which is said to be the dual of $C_0(\mathbb R^n)$. In this space we could define Fourier transform, convolution operation, ...

I am looking for a good book which is dealing with properties on $\mathcal B(\mathbb R^n)$, with all these facts about in details. So which books should I read? Do those books give the settings which we could generalize to locally compact abelian groups?

  • $\begingroup$ Did you find a good book eventually? I am looking for a similar reference which is covering Fourier transforms of finite Borel measures, convolution etc. ... $\endgroup$ – user190080 Feb 10 '17 at 17:01
  • $\begingroup$ The Fourier transform on $C_0(\mathbb{R}^n)$ seen as a (tempered) distribution ? @user190080 $\endgroup$ – reuns Feb 10 '17 at 17:05
  • $\begingroup$ terrytao.wordpress.com/2013/07/26/… $\endgroup$ – Giuseppe Negro Feb 10 '17 at 17:07
  • $\begingroup$ @user1952009 not exactly. I am looking specifically for a reference for Fourier transform on the space of finite Borel measures (I guess it also sometimes called Fourier–Stieltjes transform). $\endgroup$ – user190080 Feb 10 '17 at 17:10
  • $\begingroup$ @GiuseppeNegro so far Tao's blog actually has been my primary source...it's good, but it's not a monograph which covers this in depth. Thanks for posting it anyway. I was looking for something like Katznelson's "An introduction to harmonic analysis" but with a deeper focus on the measure side. $\endgroup$ – user190080 Feb 10 '17 at 17:16

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