$D$-optimal designs originate in statistics literature as an approach for optimal experimental designs. In numerical analysis points and weights resulting from maximal determinants turned out to be useful for quadrature and interpolation. Also recently, two of the present authors and coauthors investigated a connection to the discretization problem for the uniform norm. Here we use this approach of maximizing the determinant of a certain Gramian matrix with respect to points and weights for the construction of tight frames and exact Marcinkiewicz-Zygmund inequalities in $L_2$. We present a direct and constructive approach resulting in a discrete measure with at most $N \leq n^2+1$ atoms, which discretely and accurately subsamples the $L_2$-norm of complex-valued functions contained in a given $n$-dimensional subspace. This approach can as well be used for the reconstruction of functions from general RKHS in $L_2$ where one only has access to the most important eigenfunctions. We verifiably and deterministically construct points and weights for a weighted least squares recovery procedure and pay in the rate of convergence compared to earlier optimal, however probabilistic approaches. The general results apply to the $d$-sphere or multivariate trigonometric polynomials on $\mathbb{T}^d$ spectrally supported on arbitrary finite index sets~$I \subset \mathbb{Z}^d$. They can be discretized using at most $|I|^2-|I|+1$ points and weights. Numerical experiments indicate the sharpness of this result. As a negative result we prove that, in general, it is not possible to control the number of points in a reconstructing lattice rule only in the cardinality $|I|$ without additional condition on the structure of $I$. We support our findings with numerical experiments.

翻译：$D$-最优设计起源于统计学文献，是一种用于最优实验设计的方法。在数值分析中，由最大行列式导出的点与权重被证明对数值积分与插值具有实用价值。近期，本文两位作者与合作者进一步研究了该方法与一致范数离散化问题的关联。本文采用最大化特定格拉姆矩阵行列式（关于点与权重）的方法，在$L_2$空间中构造紧框架与精确的Marcinkiewicz-Zygmund不等式。我们提出一种直接且构造性的方法，得到一个至多包含$N \leq n^2+1$个原子的离散测度，能够对给定$n$维复值函数子空间中函数的$L_2$范数实现精确的离散子采样。该方法同样适用于从一般$L_2$空间中的再生核希尔伯特空间重构函数，其中仅需利用最重要的特征函数。我们以可验证且确定性的方式构造加权最小二乘恢复所需的点与权重，其收敛速率虽不及先前最优的概率方法，但仍具有理论保证。该通用结果适用于$d$维球面或谱支撑在任意有限指标集$I \subset \mathbb{Z}^d$上的多元三角多项式空间$\mathbb{T}^d$，其离散化最多需要$|I|^2-|I|+1$个点与权重。数值实验验证了该结果的尖锐性。作为负面结论，我们证明在缺乏$I$结构附加条件的情况下，仅通过基数$|I|$通常无法控制重构格点规则所需的点数。数值实验进一步支持了我们的结论。