On a problem of optimal recovery for integrals in the sense of Cauchy's principal value and Hadamard's finite value (Q1390368)

For linear operators \(I: X\to Y\) and \(S: X\to Z\) where \(X\) denotes a vector space and \(Y,Z\) are normed spaces, consider the problem of recovering \(Sf\) from perturbed values \(y\in Y \) of \(If\), where \(f\) belongs to a source set \(K\subset X\) and \(\| If-y\|\leq \varepsilon\) is satisfied for some \(\varepsilon \geq 0\). The associated worst case error \(E_\alpha(K,\varepsilon)\) of an algorithm \(\alpha: Y\to Z\) and the best possible worst case error \(E(K,\varepsilon)\) with respect to all algorithms are defined as follows then, \[ E_\alpha(K,\varepsilon)= \sup_{f\in K, y \in Y, \| If-y\|\leq \varepsilon} \| Sf- \alpha(y)\|, \qquad E(K,\varepsilon)= \inf_{\alpha} E_\alpha(K,\varepsilon), \] and an algorithm \(\alpha_*: Y \to Z\) is called optimal if \(E_{\alpha_*}(K,\varepsilon)= E(K,\varepsilon)\) holds. The author considers the specific situation where \(X\) is the space of functions \(f:[-1,1] \to \mathbb{R}\) with absolute continuous derivates up to the order \(n-1\geq 0\), the space \(Y= \mathbb{R}^n \) is supplied with the maximum norm, \(Z= \mathbb{R}\), and \[ If= \Big(f(t),f'(t), \ldots, f^{(n-1)}(t)\Big), \qquad Sf= \int_{-1}^{1} \frac{f(x)}{(x-t)^m | x-t |^\lambda} dx, \tag{1} \] \[ K= \Bigg\{f \in X: \| f^{(n)} \|_{L_p(-1,t)} \leq \Big(\tfrac{1+t}{2}\Big)^{1/p}, \quad \| f^{(n)} \|_{L_p(t,1)} \leq \Big(\tfrac{1-t}{2}\Big)^{1/p} \Bigg\}, \] where \( -1< t< 1\), \(0 \leq \lambda< 1\), \(1 \leq p \leq \infty\) and the integer \(m \geq 0\) are fixed. For \(m=1\), the integral in (1) is understood in the sense of Cauchy's principal value, and for \( m \geq 2 \) it is understood in the sense of Hadamard's finite value. For this situation an optimal algorithm is considered, and the best possible worst case error is calculated for the cases \( n > m \) and \( n= m\), \(p > 1/(1-\lambda) \). The results are illustrated for several values of \( m \) and \( \lambda \).