We consider the classes of functions that are defined on a metric compact, take values in an L -space (i.e. semi-isotropic semi-linear metric space) and have a given majorant of modulus of continuity. For a wide class of operators Λ that act on such function classes, we solve the problem of the optimal recovery based on inaccurate values of the functions in a finite number of points. As an application of the obtained results, we give the solutions of the problems of optimal recovery of the solutions for operator equation of the form x = f + Λ x , and in particular for the Fredholm, Volterra, and Volterra-Fredholm integral equations of the second kind for functions with values in L -spaces. As consequences of the results in the article, one can obtain new results for optimal recovery of the operators acting in the spaces of set-valued and fuzzy-valued functions, as well as in the spaces of functions with values in Banach spaces; in particular, in the spaces of random processes.

中文翻译：

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函数 L 空间中算子的最优恢复

我们考虑定义在度量紧凑上的函数类，在 L 空间（即半各向同性的半线性度量空间）中取值并具有给定的连续性模量的主要项。对于作用于此类函数类的大量运算符 Λ，我们解决了基于有限数量点中函数的不准确值的最佳恢复问题。作为所得结果的应用，我们给出了 x = f + Λ x 形式的算子方程解的最优恢复问题的解，特别是对于 Fredholm、Volterra 和 Volterra-Fredholm 积分方程的解第二种用于在 L 空间中具有值的函数。作为文章中结果的结果，可以获得新的结果，用于在集合值和模糊值函数空间中以及在具有 Banach 空间中的值的函数空间中起作用的算子的最优恢复；特别是在随机过程的空间中。