Fourier criterion for descent under a torus covering

Statement

A smooth function $f$ on $\mathbb T^n$ descends through $p_A$ precisely when

\widehat f(m)=0\qquad(m\notin A^T\mathbb Z^n).

Here $A$ is a nonsingular integer matrix.

Proof and operations preserving descent

If $f=F\circ p_A$ , pullback sends frequency $k$ to $A^Tk$ . Conversely, if the stated support condition holds, set $\widehat F(k)=\widehat f(A^Tk)$ ; these coefficients decay faster than every power and reconstruct a smooth $F$ with $f=F\circ p_A$ .

The same criterion follows from deck invariance: a frequency $m$ is invariant under every translation $A^{-1}z$ , $z\in\mathbb Z^n$ , exactly when $A^{-T}m\in\mathbb Z^n$ . Translations, averaging, and Fourier multipliers preserve this frequency lattice whenever the multiplier defines a smooth output. This also applies to a directional Fourier inverse when its denominators obey a suitable lower bound. These operations preserve descent, not arbitrary support in the auxiliary variable.