Suppose `M` is a piecewise constant function on an interval `T` taking values +1 and -1, and that `M` exhibits all the properties sufficient to ensure the existence of some converging Fourier series decomposition on `T`. Make no assumptions about the evenness or oddness of `M`, merely that all discontinuities of `M` on `T` occur where `M` changes sign. 

Write `F(M)` for the presentation of `M` as a converging Fourier series on T. Write `f(M)` for a 'low-passs' filtered `F(M)`, i.e. suppose `f(M)` to be `F(M)` where all the terms in `F(M)` having frequencies above some fixed predetermined value have been removed from the sum (so `f(M)` is some partial sum). 

What is the relationship between the set of zeros for `F(M)` and `f(M)`?