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Suppose there is a distribution that has values sampled on the interval 1-25 with corresponding sample values that have to be smoothed. For example:

n    f[n]

3    123
5    234
8    246
10   567
14   253
18   251
24   132

The first column has sample points between 1-25 but does not have samples for all points on the interval. Should interpolation of other points in the distribution be done so that we have approximate values for all points between 1-25 before applying Gaussian smoothing?

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Irregularly sampled data need not necessarily be interpolated prior to smoothing. You can try to smooth your data in the following way:

$$y(t_n)=\frac{\sum_{k}g(t_n-t_k)x(t_k)}{\sum_{k}g(t_n-t_k)}\tag{1}$$

where $t_n$ are your sample times, $x(t_n)$ are your data, and $g(n)$ is your smoothing kernel. Note that $t_n$ need to be integers (as in your example), which is no restriction, just a matter of formulation. Since $t_n$ are irregular, you do not always use all values of the chosen kernel $g(n)$, but this is compensated for by the normalization in (1). The kernel width has to be chosen such that the kernel always covers at least several of your irregularly sampled data. The smoothing in (1) is also used in the Nadaraya-Watson estimator.

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