What is the underlying function of 'interpolate' option of Volumes module

I have a resampled volume which looks smoother when the interpolate option is checked. I would like to save the volume like that. What is the actual function it uses to do that?

That option controls how the image is sampled when scaled for display: nearest neighbor / linear (see https://en.wikipedia.org/wiki/Image_scaling for details). There is nothing to save - what you see is still the original volume.