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 $f(M)$?

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. $f(M)$ is $F(M)$ where all the terms in $F(M)$ having frequencies above some fixed predetermined value have been removed from the sum (so $f(M)$ is a partial sum).

What is the relationship between the zero sets of $F(M)$ and $f(M)$?