Let us determine the effect of the filter on the Fourier coefficient . We start with the coefficient of , which is given by
where we arbitrarily chose as the periodic interval. The Fourier coefficient of is similarly given by
where we used the definition of in terms of . We now make the change of variables on the inner integral, implying and , leading to
Since the integral on is over the whole period, we may now change variables on it without changing the integration limits, using , with and , and thus obtaining
were we recognized the form of . In this way we show that , that is, the filter does not change at all. Another way to state this is to say that the filter does not change the average value of over the periodic interval.