In a Hilbert space defined as the image of a unit ball under the action of a compact operator, we solve the problems of optimal recovery of elements according to their first n Fourier coefficients known with errors. Similar problems are also solved for the scalar products of elements from two different classes.

中文翻译：

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根据误差已知的傅立叶系数，对希尔伯特空间中的元素及其标量积进行最佳恢复

在希尔伯特空间中，定义为单位球在紧致算子的作用下，我们根据已知的具有误差的前n个傅立叶系数解决了元素的最佳恢复问题。对于来自两个不同类别的元素的标量积，也解决了类似的问题。