I have a function $f :\mathbf{R}^2 \to \mathbf{R}$ under different discretizations, e.g., $64 \times 64$ and $32 \times 32$ (can be viewed as a $64 \times 64$ image downsampled to $32 \times 32$). Now, I Fourier-transform both, obtain two complex matrices of sizes $256 \times 256$ and $64 \times 64$, respectively, call them matrices $A$ and $B$. Assume low-frequency dominance, i.e., the discretizations are mainly of smooth changes, no sharp edges etc.
I multiply some random weights to the low-frequency components of $A$, and the same weights to the same low-frequency components of $B$, then do inverse transform on both results. What I observed is that the resulting images looks very similar, in fact, if I downsample the $64$ one to $32$, the results look almost identical. Is there any mathematical reasoning behind this?
