definition of Fourier coefficient for map of vectorbundles I am trying to understand the elementary proof of atiyah and bott for bott periodicity(cf. Atiyah's book K-Theory, page 51). In the proof they define fourier coefficients, but I can't see how their definition makes sense.
Let $X$ be a space and let $L$ be a line bundle over $X$. Choose a metric on $L$ and let $S \subset L$ be the unit circle bundle and denote by $\pi: S \rightarrow X$ the projection map. Let $z : S \rightarrow \pi ^*(L)$ be the section given by the diagonal map $s \mapsto (s,s) \in L \times_XS = \pi ^*(L)$. If $E_1,E_2$ are bundles over $X$ and $f\in \Gamma \text{Hom}(\pi ^* E_1, \pi ^* E_2)$ then atiyah and bott define the k'th Fourier coefficient $a_k\in \Gamma \text{Hom}(L^k \otimes E_1, E_2) $ of $f$ as 
$$a_k(x) = \frac{1}{2\pi i} \int_{S_x} f_x \otimes z_x^{-k-1} \text{d}z_x$$ 
But i can't even make sense of $f_x \otimes z_x^{-k-1}$, since $z$ is a section into $\pi ^*(L)$ and the $f$ is a section into a Hom vectorbundle. How am I supposed to tensor them and how is $a_k$ real a section into $\text{Hom}(L^k \otimes E_1, E_2) $?
 A: The section $f$ is of the bundle
$$
\operatorname{Hom}(\pi^*E_1,\pi^*E_2)
= \pi^* \operatorname{Hom}(E_1, E_2)
$$
over the manifold $S$ (not $X$, nota bene). We can tensor this with a power of the section $z$ of the bundle $\pi^*S$ over $S$ (not $X$), and get a section of 
$$
\pi^*(S^{k} \otimes \operatorname{Hom}(E_1,E_2)) 
\subset \pi^*(L^{k} \otimes \operatorname{Hom}(E_1,E_2))
$$ 
over $S$. We can then take the pushforward of this section under $\pi$ and get a section of 
$$
L^{k} \otimes \operatorname{Hom}(E_1,E_2) 
= \operatorname{Hom}(L^{-k} \otimes E_1,E_2)
$$ 
over $X$. The pushforward comes down to exactly the integral over each unit circle. It takes a little work to figure out the details (working out the case where $X$ is a point helps), but it's good practice. (Question 1: Why is the change in sign $k \to -k$ is correct? Question 2: But where does the extra $-1$ in $z^{-k-1}$ come from, Michael? (Hints: Michael - see Arrested Development. For the $-1$, think about the Lebesgue measure on the unit circle from complex analysis.))
It could be worth your time to spend a little while thinking about pullbacks of vector bundles (things like $p^*E \to E$ where $p : E \to X$ is a vector bundle) and the various canonical isomorphisms of vector spaces (like $L^k \otimes \operatorname{Hom}(V,W) = \operatorname{Hom}(L^{-k} \otimes V, W)$). It's not particularly complicated stuff, but it takes a while to get used to it. Always remember: the simplest manifold is a point, and a vector space is just a vector bundle over a point. That'll get you plenty of cheap simple examples to practice on.
