Nonnegative or positive band-limited interpolation
If we require the impulse response of the interpolation filter to be non-negative then we can obviously guarantee that the interpolated signal is also non-negative. However, it can be shown that the frequency response of a low pass filter with non-negative impulse response is always far from ideal. More specifically, it can be shown that if $\epsilon$ is the maximum negative pass band deviation of the squared magnitude $|H(\omega)|^2$ of the filter's transfer function from its value at $\omega=0$

$$|H(\omega)|^2\ge |H(0)|^2-\epsilon,\quad |\omega|&lt;\omega_c \tag{1}$$

where $\omega_c$ is the cut-off frequency, then the following inequality holds:

$$|H(\omega)|^2\ge |H(0)|^2-4\epsilon,\quad |\omega|&lt;2\omega_c\tag{2}$$

This inequality imposes a severe restriction on the minimum stop band attenuation if the ripple size $\epsilon$ is chosen sufficiently small (as should be the case for a good low pass filter).

As an example, if $\epsilon$ is chosen such that the pass band ripple is $1$ dB, the attenuation at $\omega=2\omega_c$ cannot be better than $7.5$ dB.

Of course, if $\epsilon\ge |H(0)|^2/4$ then $(2)$ represents no restriction, but in that case the maximum pass band deviation of the magnitude of the transfer function is greater than usually required for a good low pass filter.

Since the inequality $(2)$ is independent of the filter order, an interpolation filter with arbitrary accuracy cannot be realized by a filter with non-negative impulse response.

The proof of $(2)$ can be found in Attenuation Limits for Filters with Monotonic Step Response by A. Papoulis, IRE Transactions on Circuit Theory ( Volume: 9, Issue: 1, March 1962).