Under investigation is the problem of finding the best approximation of a function in a Hilbert space subject to convex constraints and prescribed nonlinear transformations. We show that in many instances these prescriptions can be represented using firmly nonexpansive operators, even when the original observation process is discontinuous. The proposed framework thus captures a large body of classical and contemporary best approximation problems arising in areas such as harmonic analysis, statistics, interpolation theory, and signal processing. The resulting problem is recast in terms of a common fixed point problem and solved with a new block-iterative algorithm that features approximate projections onto the individual sets as well as an extrapolated relaxation scheme that exploits the possible presence of affine constraints. A numerical application to signal recovery is demonstrated.

正在研究的问题是在受凸约束和规定的非线性变换的 Hilbert 空间中找到函数的最佳近似值。我们表明，在许多情况下，即使原始观察过程是不连续的，也可以使用牢固的非膨胀算子来表示这些处方。因此，所提出的框架捕获了谐波分析、统计、插值理论和信号处理等领域中出现的大量经典和当代最佳近似问题。由此产生的问题根据一个常见的不动点问题重新定义，并使用一种新的块迭代算法解决，该算法具有对单个集合的近似投影以及利用仿射约束可能存在的外推松弛方案。