Function-Theoretic and Probabilistic Approaches to the Problem of Recovering Functions from Korobov Classes in the Lebesgue Metric

In this article, function-theoretic and probabilistic approaches to the recovery of functions from Korobov classes in Lebesgue metrics are considered. Exact order estimates are obtained for the recovery errors of functions reconstructed from both accurate and inaccurate information given by the trigonometric Fourier–Lebesgue coefficients of the recovered function in the uniform metric. Within these settings, optimal computational aggregates (optimal recovery methods) are constructed. The boundary of inaccurate information (the limiting error ε˜N) that preserves the order of recovery corresponding to accurate information is identified. Furthermore, a set of computational aggregates is constructed whose limiting errors do not exceed ε˜N. A procedure for constructing a probability measure on functional classes is presented, and upper bounds for the mean-square recovery error with respect to these measures on Korobov classes are established. Numerical experiments were conducted to validate the theoretical results. These experiments showed that for the function corresponding to the lower bound in Theorem 1 (cases C(N)D-2 and C(N)D-3), the ratio between the function value and the approximation error remains constant in the case of uniform weighting and increases indefinitely when logarithmic weighting is used as the number of terms N grows.