Denseness of the spaces ${\displaystyle {\Phi }_V}$ of Lizorkin type in the mixed ${\displaystyle L^{p̅}\left({ℝ}^n\right)}$-spaces

The spaces Φ_V(ℝ^{n}) are defined to consist of Schwartz test functions φ such that the Fourier transform φ̂ and all its derivatives vanish on a given closed set V ⊂ ℝ^{n}. Under the only assumption that m(V) = 0 it is shown that Φ_V is dense in ${\displaystyle C_0\left({ℝ}^n\right)}$ and in the space ${\displaystyle L^{p̅}\left({ℝ}^n\right)}$ with the mixed norm, for ${\displaystyle \frac{1}{p}̅}$ in a certain pyramid. The result on the denseness for arbitrary ${\displaystyle p̅=(p_1,...,p_n)}$, 1 < p_k < ∞, k = 1,...,n,!$! is proved for so-called quasibroken sets V.