Optimal weighted sampling of positive and normal operator orbits

We study weighted sampling of operator orbit frames in infinite dimensional separable Hilbert spaces $H$, keeping the operator and generating vectors fixed. Let $N_{\mu }(R)$ count the sampling times in $[0,R]$. For a single orbit of a positive operator, let $m_A(R)$ count the eigenvalues in $(0,e^{-1\mathrm{/}R}]$. Every real sampling design that preserves the frame property must satisfy the lower bound $N_{\mu }(R)\ge m_A(R)-O_{\mu }(1)$. Conversely, for every $\eta >0$, we construct an integer sampling design that preserves the frame property and satisfies $N_{\mu }(R)\le (1+\eta )m_A(R)+O_{\eta }(1)$. Thus the coefficient $1$ is optimal. Under additional spectral conditions, we attain $N_{\mu }(R)=m_A(R)+O(1)$ and characterize the weights that preserve the frame property on the constructed sets. For finitely or countably many orbits of a positive operator, let $P_j$, $j\ge 1$, be the spectral projection corresponding to $2^{-j-1}<-\log \lambda \le 2^{-j}$ and define $K_A(R)=\mathrm{\#}\{j\ge 1:2^j\le R,P_j\mathrm{\ne }0\}$. Every real sampling design that preserves the frame property satisfies $N_{\mu }(R)\ge c_{\mu }{K}_A(R)-C_{\mu }$ for $R\ge 2$, with constants $c_{\mu }>0$ and $C_{\mu }\ge 0$. Conversely, we construct an integer sampling design that preserves the frame property and satisfies $N_{\mu }(R)\le L{K}_A(R)+L$ for $R\ge 2$, where $L$ depends only on the bounds of the given orbit frame. For normal operators of norm one, we prove that the smallest frame condition number is attained among integer sampling designs that preserve the frame property and satisfy a prescribed counting bound, whenever this class is nonempty. Finally, we construct one normal operator with two families of generating vectors for which the optimal growth orders of $N_{\mu }(R)$ are $\log R$ and $R$, respectively, for both real and integer sampling.