On Optimal Recovery and Information Complexity in Numerical Differentiation and Summation

TL;DR

This paper investigates the limits of accuracy and information requirements for numerical differentiation and summation, providing sharp error estimates and algorithms, and analyzing conditions for problem well-posedness.

Contribution

It offers new sharp estimates of optimal recovery errors and information complexity, along with algorithms based on truncation and Chebyshev polynomials for these problems.

Findings

Sharp order estimates of recovery error and information complexity.

Conditions under which summation is well-posed.

Algorithms for numerical differentiation and summation based on Chebyshev polynomials.

Abstract

In this paper, we study optimization problems of numerical differentiation and summation methods on classes of univariate functions. Sharp estimates (in order) of the optimal recovery error and information complexity are calculated for these classes. Algorithms are constructed based on the truncation method and Chebyshev polynomials to implement these estimates. Moreover, we establish under what conditions the summation problem is well-posed.