Representing a positive 0-limiting continuous time domain function, in terms of spectral samples, outside of a window
Finite $X_k$ samples will inevitably lead to a periodic approximation. The question, as is, is unfortunately and quite interestingly, unanswerable.

Indeed, under the standard formalisms, the reconstructed signal $\hat x(t)$ will be peridic:
$$
F{X(f)|||(f)}=\sum^{\inf}_{i=-\inf} X(i\Delta f)e^{i2\pi i\Delta f t}\Delta f = \hat x(t)
$$

If we truncate the signal $x(t)$ to be time-bounded, and using a similar argument than the Paley Wiener theorem to the direct Fourier Transform, we will obtain a exponentially converging spectrum on $X(f)$ when $f$ tends to infinity.

Based on the last, a logarithmic approximation could be sketched, in which the $f_i$ are being preferred when closest to zero, and sparse for increasing values:
$$
\hat x(t)=\sum^{\inf}_{f_i=-\inf} X(f_i)e^{i2\pi f_i t}\Delta f_i
$$

And under such aproximation, we could let the first terms of the sum to be again, approximated with known spectrum shapes (as $sinc(f)$ as you said). But that is not a train of frequency samples. This is a different work, and perhaps a different question?.