Take a look at the following diagrams:
The original picture:
The upscaled (2x, bilinear) picture:
In both cases the first item in the first row is the input image. There is its 2d fft next to it. Below it there is a downscaled version of the image and its 2d fft. Notice the difference in spectrum amplitude in both cases. While in the first case nothing changes, in the second the spectrum actually gets better (it was weak in the first place)—I mean there is more "weight" put in the corners.
Is it some sort of well-known property? Should I expect the upscaled image to be crisper when downscaling it?