Notation for unevenly-spaced to evenly-spaced time series conversion I have an unevenly-spaced time series. To make it evenly-spaced, I resample the time series to a larger timespan (e.g. day) and sum up all values within this time frame. 
In Python (and pandas) it is a one-liner:
ts.resample('D').sum()

But what is a precise (and ideally easily unterstandable) mathematical notation for that?
 A: Let's start with a generalization.  Suppose the original data are a series of values $\mathbf{x}=(x_1, x_2, \ldots, x_n)$ associated with times $\mathbf{t}=(t_1, t_2, \ldots, t_n).$  You have in mind a new set of ordered times indexed by the integers $\mathbf Z$ along with a function $f(k, \mathbf{t}, \mathbf{x})$ that determines what new value to assign to the integer $k$ based on the data in $(\mathbf{t}, \mathbf{x}).$  The new time series is just
$$(y_k) = (f(k, \mathbf{t}, \mathbf{x})),\ k\in\mathbb{Z}.$$
In the example in the question, you have partitioned an interval of time from $t_0$ to $t_\infty \gt t_0$ into (say) $N$ evenly spaced intervals, each of which therefore has a length $$h= (t_\infty - t_0)/N,$$ and I am presuming (as is conventional) that you wish to index them with the integers $0, 1, 2, \ldots, N-1.$  The interval in which any time $t$ lies therefore is
$$g(t) = \lfloor \frac{t-t_0}{h} \rfloor.$$
Your resampling function sums all values within each interval and therefore can be written
$$f(k,\mathbf{t},\mathbf{x}) = \sum_{\{i\,\mid\, g(t_i)=k\}}x_i.$$
(Empty sums are, by definition, zero.)
If you like, you may combine these formulas in one grand (but less immediate) one,

$$y_k =  \sum_{\{i\,\mid\, \lfloor N\frac{t_i-t_0}{t_\infty-t_0} \rfloor=k\}}x_i.$$

To demonstrate how non abstract and natural this is, here is an R implementation of the last formula:
function(times, x, t.0, t.infty, N) tapply(x, floor(N * (times - t.0)/(t.infty - t.0)), sum)

It returns an array indexed by the new times with values equal to the sums within each new interval.
