On Optimal Recovery and Information Complexity in Numerical Differentiation and Summation

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.

Figures (4)

• Figure 1: Recovery of the derivative $f^{(1)}_1$ with random noise in the input data. Approximation to $f^{(1)}_1$ for $\delta= 10^{-4}$ (Fig. a) ), for $\delta= 10^{-5}$ (Fig.b) ) and $\delta= 10^{-6}$ (Fig. c) )
• Figure 2: Recovery of the derivative $f^{(2)}_1$ with random noise in the input data . Approximation to $f^{(2)}_1$ for $\delta= 10^{-4}$ (Fig. a) ), for $\delta= 10^{-5}$ (Fig. b) ) and $\delta= 10^{-6}$ (Fig. c) )
• Figure 3: Summation of $f_2$ with random noise in the input data for $\delta= 10^{-2}$ (Fig. a) ), for $\delta= 10^{-3}$ (Fig. b) ) and $\delta= 10^{-4}$ (Fig. c) )
• Figure 4: Recovery of the derivative $f_3^{(2)}$ with noise from Clenshaw-Curtis quadrature. Approximation to $f_3^{(2)}$ for $\delta= 10^{-5}$ (Fig. a)), $\delta= 10^{-6}$ (Fig. b)) and $\delta= 10^{-7}$ (Fig. c))

Theorems & Definitions (44)

• Remark 1.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• proof
• Corollary 2.4
• Lemma 3.1
• proof