Optimal approximation of periodic analytic functions with integrable boundary values
The main result of this paper is the next Theorem 1. Suppose that \(1\leq p\leq q< \infty\) or that \(q= 2\) and \(p\) satisfies \({1\over p}> {2\over \pi} \sum^\infty_{m=0} {{(-1)^m} \over {2m+1}} {1\over {\cosh (2m+1) \beta}}\). Then \[ d_{2n} (A^q, L_p)= d^{2n} (A^q, L_p)= \delta_{2n} (A^q, L_p)= \inf_{B\in {\mathcal B}_{2n}} \sup_{g\in A^q} |qB |_{L_p}. \] Moreover, sampling is optimal for \(\delta_{2n} (A^q, L_p)\). Here \(d_n\), \(d^n\) and \(\delta_n\) is Kolmogorov, Gel'fand and linear \(n\)-widths of a unit ball \(A^q\) of a space \(H^q (\Omega)\) \((\Omega= \{z\in \mathbb{C}: R<|z|< {1\over R}\}\), \(R= e^{-\beta}\), \(\beta>0)\), which consider in the corresponding Banach space \(L_p\), and \({\mathcal B}_{2n}\) is the set of all Blaschke products of degree \(2n\) all whose zeros lie on the unit circle in \(\mathbb{C}\).
scientific article; zbMATH DE number 847527