I am learning about Riemann surfaces, using the books of Forster and Miranda. When stating and proving Riemann-Roch and Serre Duality in both books a space appears which seem somehow related, but this relation is not immediately clear (to me).
These spaces are the following:
Let $X$ be a compact Riemann surface.
In Miranda's book $$H^1(D)= \tau [D](X)/\operatorname {Im}(\alpha_D),$$ where $\tau[D](X)$ is the group of Laurent tail divisors on $X$ corresponding to the Divisor $D$ and $\alpha_D$ is the truncation map which removes all terms of order $-D(p)$ or higher from the Laurent series of a meromorphic function. I.e. $H^1(D)$ measures the failure of being able to solve Mittag-Leffler problems on $X$ of finding a meromorphic function with a given tail.
In Forsters book the space of interest is the first cohomology group $H^1(X,\mathcal O_D)$ with $\mathcal O_D$ the sheaf of meromorphic functions $f$ with $\operatorname{div} f \ge -D$.
With $\Omega^1_D$ the sheaf of meromorphic 1-forms $\omega$ satisfying $\operatorname{div} \omega \ge -D$.
Serre duality then reads as
$$ H^1(D)= \Omega^1_{-D}(X),~~~ \text{ resp } ~~~H^1(X,\mathcal O_D)= \Omega^1_{-D}(X).$$
In both cases the duality map is given by the residue map. And the proofs are somehow similar, as the strategy is to shaw that the suitable defined resiude map is an isomorphism.
In order to calculate the Residue of a cohomology class, Forster introduces so called Mittag-Leffler distributions, which for a given open covering $\mathcal U$ are cochains $\mu \in C^0(\mathcal U, \mathcal M^1)$ with $\delta \mu \in Z^1(\mathcal U, \Omega^1)$, i.e. $\mu_j- \mu_i$ is holomorphic on $U_i \cap U_j$ and defines $\operatorname{Res} (\mu)= \sum_{a \in X} \operatorname{Res}_a (\mu)$.
He then proves $\operatorname{Res}([\delta \mu])= \operatorname{Res}(\mu)$.
I got the feeling, that both authors do basically the same, but use different words to describe it. Using both versions of Serre duality, the spaces are isomorphic, but there seems to be a more direct connection.
So what is it? In particular, how are Mittag-Leffler Distributions related to Mittag-Leffler problems? So far I could only observe that for a Mittag-Leffler distribution the Laurent series of the $\mu_i$ have the same principal part.
The bounty ended and unfortunately the question was not answered satisfactorily, even a year later. Hopefully somebody can step in.
$\def\CC{\mathbb{C}}\def\cO{\mathcal{O}}\def\cK{\mathcal{K}}$There are three definitions of coherent sheaf cohomology I'll want to compare here:
(1) Miranda's definition (which is pretty close to the adelic definition of Weil).
(2) Sheaf cohomology of algebraic functions in the Zariski topology.
(3) Sheaf cohomology of holomorphic functions in the analytic topology (Forster's definition).
The equality of (2) and (3) is Serre's GAGA, and I'm not sure I can say anything insightful about it. This answer will focus on the equivalence of (1) and (2).
Let $D$ be a divisor supported on a finite subset $S$ of $X$. Let $U$ be the open set $X \setminus S$. Let $\cO_U$ be the sheaf on $X$ obtained by pushing forward the structure sheaf $\cO$ from $U$. Note that $\cO_U$ is acyclic, since it is pushed forward from the affine variety $U$. For $s \in S$, let $(\cO_s)_s$ be the skyscraper sheaf whose value at $s$ is the local ring $\cO_s$. Let $\cK$ be the field of meromorphic functions on $X$ and let $\cK_s$ be the skyscraper sheaf equal to $\cK$ at $s$. Note that, since they are skyscraper sheaves, $(\cO_s)_s$ and $\cK_s$ are acylic.
Then we have a short exact sequence of sheaves: $$0 \to \cO \to \cO_U \oplus \bigoplus_{s \in S} (\cO_s)_s \to \bigoplus_{s \in S} \cK_s \to 0.$$ Exactness can be easily checked on stalks. So $H^1(X, \cO)$ is the cokernel of $$H^0(X \setminus S, \cO) \oplus \bigoplus_{s \in S} \cO_s \longrightarrow \bigoplus_{s \in S} \cK.$$ We can quotient by the image of the second summand, and get that $H^1(X, \cO)$ is the cokernel of $$H^0(X \setminus S, \cO) \longrightarrow \bigoplus_{s \in S} \cK/\cO_s.$$
The quotient $\bigoplus_{s \in S} \cK/\cO_s$ is finite sums $\sum^{\mathrm{finite}}_{j < 0} c_j z_s^j$ where $z_s$ is a coordinate near $s$. So $$H^1(X, \cO) \cong \mathrm{CoKer} \left(H^0(X \setminus S, \cO) \longrightarrow \bigoplus_{s \in S} \left\{ \sum\nolimits^{\mathrm{finite}}_{j < 0} c_j z_s^j \right\} \right).$$
If we tensor everything with $\cO(D)$, we similarly get $$H^1(X, \cO) \cong \mathrm{CoKer} \left(H^0(X \setminus S, \cO) \longrightarrow \bigoplus_{s \in S} \left\{ \sum\nolimits^{\mathrm{finite}}_{j < -D_s} c_j z_s^j \right\} \right).$$
Now, this still isn't quite Miranda's definition. The meromorphic functions on $X$ are $\lim\limits_{S \to} H^0(X \setminus S, \cO)$ and the Laurent tails are the limit of the right hand side of $S$. So Miranda is working with $$\mathrm{CoKer} \left( \lim\limits_{S \to}H^0(X \setminus S, \cO) \longrightarrow \lim\limits_{S \to} \bigoplus_{s \in S} \left\{ \sum\nolimits^{\mathrm{finite}}_{j < -D_s} c_j z_s^j \right\} \right).$$
Our previous computation shows that the cokernel stabilizes at $H^1(X, \cO_D)$ once $S$ is large enough to support $D$, and cohomology commutes with forward limits, so Miranda's definition works.