Suppose I have $P$, which is the base-$p$ representation of an integer $n$ and I want to calculate it base-$q$ representation $Q$. The obvious algorithm is: interpret $P$ to obtain $n$, then calculate $Q$ from $n$. This is easy.
But if $n$ is very large this could require a lot of memory, and it could require a large amount of processing before any of the digits of $Q$ are emitted. In certain special cases we can do better; there is an online algorithm which can start producing output before all the input is read. For example, if $p^i = q^j$ then the algorithm can read $i$ digits of $P$ and immediately produce $j$ digits of $Q$ without any further input and with only a constant amount of memory. This method is used, for example, in converting generic computer data ($p=256$) to “Base64” format ($q=64$).
Is there a general online algorithm for radix conversion? I expect that if there is such an algorithm, in the worst case it cannot emit any digits of $Q$ without first reading all of $P$, but are there algorithms that can often do better than this, in cases more general than the very unusual $p^i=q^j$ situation? Or are there any other special cases of interest?
[ Related thread: The math behind converting from any base to any base without going through base 10? but all of the suggestions there ingest the entire input before producing any output. ]