Let $(t_k), k \in \mathbb{N}$, be an increasing sequence of real numbers ($t_{k-1} < t_k$) and $(X_{t_k}$) be a sequence of real numbers indexed by $(t_k)$. Such a sequence is sometimes called a time series.

The idea is that this series represents a sequence of measurements of some sort, like, for example, the average temperature of some location at time $t_k$.

The analysis of time series is an established area of statistics. In concrete applications, for example in climate science, there are two common problems when applying statistical algorithms to time series:

The time series are finite, which produces artefacts in statistical algorithms that are designed for infinite time series. This problem is well known and there exist several approches to handle it.

The times series are uneven spaced, that is $t_k - t_{k-1}$ is not independent of $k$.

I don't know of any textbook, algorithm or paper that explicitly addresses the latter problem. My question is therefore: Is this not a problem, is the solution trivial or, if not, are there any treatments?

Of course it is possible to interpolate missing values to generate a time series with an even time spacing $\min_k (t_k - t_{k-1})$, but it seems to me that this is not a solution, because algorithms like the fast fourier transform, nonlinear regression analysis or wavelet transforms would produce artefacts that depend on the kind of interpolation (linear, qubic splines, whatever). And therefore an explicit explanation of why the kind of interpolation one uses does not produce any artefacts in the analysis of the time series seems to be warranted to me, but I have never seen one in the literature.