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I'm currently working with Fourier transforms of measures on the $\mathbb{T}^n$ (more specifically in dimension two), i.e. $$ \hat{\mu}(k) = \int_{\mathbb{T}^n} e^{i k \cdot x} d\mu(x) $$ or something of that form. I am unfamiliar with this theory and would really appreciate a good reference on this topic.

Would anyone be able to point me to a good reference on the Fourier transform of measures over some unit cell? I have found literature for when $\mathbb{T}$ is replaced with $\mathbb{R}$, but am struggling to find a good reference for the requested case.

In case it is relevant, I am interested in the case when $k$ takes values on some lattice.

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    $\begingroup$ Since you want $\mathbb T$, not $\mathbb R$, you should look up Fourier series rather than Fourier transform. $\endgroup$
    – Andreas Blass
    Commented Jul 30, 2021 at 23:31
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    $\begingroup$ @AndreasBlass Ah yes. I just thought that the general phrase transform is used when working with measures. Thank you for the hint :) $\endgroup$
    – spaceman
    Commented Jul 31, 2021 at 7:00

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You can refer to Chapter Four of

G. B. Folland, A Course in Abstract Harmonic Analysis. CRC Press, 1995.

The book really deserves the word "course" in its title.

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