When you look at a filter structure, you don't find much big mathematical operations happening. Maximum you can find a delay and a signed adder in it. Whatever may be the number of frequency components available in the input signal to the filter, the coefficients proficiently decide what values of the individual frequency components should pass out through the filter and what should die out.
the coefficients play a large role in the filter design. Any FIR filter can be normalized (not including quantization issues). stable IIR filters can also be normalized (and excluding quantization issues).