We find explicit expressions for optimal recovery methods in the problem of recovering the values of continuous linear operators on a Sobolev function class from the following information: The Fourier transform of functions is known approximately on some measurable subset of the finite-dimensional space on which the functions are defined. As corollaries, we obtain optimal methods for recovering the solution to the heat equation and solving the Dirichlet problem for a half-space.

中文翻译：

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基于一类函数谱近似信息的一类算子的最优恢复

我们从以下信息中找到了在恢复 Sobolev 函数类上的连续线性算子的值的问题中最优恢复方法的显式表达式： 函数的傅里叶变换在有限维空间的某个可测量子集上近似已知，在该子集上函数被定义。作为推论，我们获得了恢复热方程解和求解半空间狄利克雷问题的最佳方法。