One can build an Eisenstein series $E_s(v,g)$ from a vector $v\in D_s$, the principal series representation of $G_{\mathbb{A}_\mathbb{Q}}$. The space $D_s$ has a restricted tensor product structure $$ D_s=\hat{\otimes}D_{s,p}.$$ Now we assume $v$ has unramified standard finite components for $p\ne\infty$. We only consider $v_\infty=f\in D_{s,\infty}$. We know that it has a realization as functions on the unit circle. When $f$ is smooth, we have some theory on the corresponding Eisenstein series (meromorphic continuation, convergence, etc).
My question is: What can we say about (or how do we study) the Eisenstein seires if $f$ is a "more general function"? Say what if $f$ have a pole somewhere on $\mathbb{S}^1$, or $f$ is a generalized function? Does the same construction of $E_s(v,g)$ go through?
