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I know that if a function $f(x_1, x_2,\ldots)$ is periodic along a certain coordinate $x_k$: $$f(\ldots, x_k,\ldots) = f(\ldots, x_k + L_k,\ldots)\quad,$$ it leads to the discrete Fourier series along this coordinate, and so, to the discrete spectrum.

Now, what are the consequences of the function being an angle-like itself? $$f(x_1,x_2,\ldots) \sim f(x_1,x_2,\ldots)+2\pi R$$ Or, better say: $$f:\: \mathbb{R}^m \times S^n \to S^1$$

(meaning that $n$ coordinates are periodic, while $m$ are not)

A more general question — what if the function takes values in a compact space? (In Physics, we are normally interested in an even more particular case — when such a space is a manifold of a compact Lie group.)

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    The notation $f(x_1,x_2,\ldots)=f(x_1,x_2,\ldots)+R$ doesn't make sense. It's better to simply write $f:\mathbb{R}^n\to S^1$ where $S^1$ is the unit circle. – 2017-01-25
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    Please check now. – 2017-01-25

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To look at some basic features of such functions we can consider: $$f:S^1_r \to S^1_R\,, \qquad f(x+2\pi r) = f(x)\,, \qquad f(x) \sim f(x)+2\pi R,$$ where the subscripts $r$ and $R$ denote the radii of the circles. But now notice that we don't have to impose $f(x+2\pi r) = f(x)$, instead, we can relax it a bit and only impose: $$f(x+2\pi r) = f(x) + 2w\pi R\,, \qquad w \in \mathbb Z\,, \tag{S} $$ Since $f(x)$ and $f(x) + 2w\pi R$ are being identified. This leads to a Fourier series as follows: $$f(x) = w \frac{R}{r}x+\sum_{n \in \mathbb Z} a_n e^{inx/r}\,.$$ The only difference with the Fourier series of functions with noncompact codomain is the linear term which is now allowed because this only introduces a shift by $2w\pi R$ in the function as we shift $x$ by $2\pi r$, and we are identifying such shifts in (S). The integer $w$ is called a winding number, because the interpretation of this number is "the number of times the domain $S^1$ is wrapped or winded around the codomain $S^1$ under the map $f$ ".

For more general compact spaces such functions are parametrized by, roughly speaking, "in how many ways the (domain) compact space can be wrapped around the (codomain) compact space". Different ways of wrapping are related to various homotopy groups. Fourier transforms in such cases can be fairly complicated, I don't know how to write them in general, though the basic idea appears in the $S^1$ case, we must allow all terms that can cause the shifts in function which we are modding out.

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    Thanks for the response! This all makes a lot of sense, and coincides precisely with the theory of the bosonic string. It took me some time to think over the idea of the linear term... Do you have an idea about an analogous construction for the case $\mathbb{R} \to S^1$? My guess is that case is identical to $\mathbb{R} \to \mathbb{R}$. Technically, we could add a linear term with $\omega$ being an integer. But this possibility is already covered by the continuous integral in the frequency space. – 2017-01-25
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    @mavzolej You're welcome! and, I think you're right, the $\mathbb R \to S^1$ case is similar to the $\mathbb R \to \mathbb R$ case. It seems that it's more explicit if we introduce radii for the circles (which I have done in the recent edit). Now if we take the radius of the domain $S^1$ to go to infinity (i.e. $r \to \infty$) we loose the linear term and the rest becomes the integral Fourier transform of the $\mathbb R \to \mathbb R$ case. – 2017-01-25