If I have, for example, two finite lists of sine waves and all the sine waves in each of these lists are summed to create 2 new sine waves a and b, then is there any way I can figure out whether or not the 2 lists of sine waves contained any common sine waves that appeared in both lists without the underlying data (the sine waves in the lists) and only the sine waves a and b?
For Example, if I have two lists containing a finite number of sine waves: (sin(x), sin(2x), sin(4x), sin(7x)) and (sin(3x), sin(6x), sin(5x), sin(7x)), and all the sine waves in each list are summed to create two new sine waves a and b: sin(1x) + sin(2x) + sin(4x) + sin(7x) = sine wave a, and sin(3x) + sin(6x) + sin(5x) + sin(7x) = sin wave b, then is is possible to figure out that sin(7x) appears in both lists without knowing what the sine waves in the lists were and only knowing the sine waves a and b?