Approximation of restrictions of $q$-valued logic functions to linear manifolds by affine analogues

For a finite $q$-element field $\mathbf{F}_q$, we established a relation between parameters characterizing the measure of affine approximation of a $q$-valued logic function and similar parameters for its restrictions to linear manifolds. For $q>2$, an analogue of the Parseval identity with respect to these parameters is proved, which implies the meaningful upper estimates $q^{n-1}(q-1) - q^{n/2-1}$ and $q^{r-1}(q - 1) - q^{r/2-1}$, for the nonlinearity of an $n$-place $q$-valued logic function and of its restrictions to manifolds of dimension $r$. Estimates characterizing the distribution of nonlinearity on manifolds of fixed dimension are obtained.