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)?