## Optimal Recovery of Analytic FunctionsThis book is devoted to the problems of optimal recovery on classes of analytic functions. Considered are the problems of optimal interpolation, differentiation, and quadrature formulas. Many problems of approximation theory may be considered as optimal recovery problems. Those problems are Kolmogorov's inequality and n-widths which are studied for the Hardy classes. |

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### Contents

5 | |

11 | |

14 | |

13 Optimal Recovery of Linear Functionals | 20 |

14 Optimal Algorithms Using Fourier Coefficients | 25 |

15 Notes and References | 34 |

Optimal Recovery in Hp | 37 |

22 The Periodic Case | 40 |

36 Notes and References | 121 |

Exact nWidths of Analytic Functions | 123 |

41 Exact nWidths in Hp | 124 |

42 Estimates of nWidths for Hp in Lq11 | 131 |

43 Exact nWidths of Periodic Functions | 139 |

44 Estimations of nWidths in Hilbert Spaces | 146 |

45 Diagonal Operators | 155 |

46 HardySobolev and BergmanSobolev Classes | 161 |

23 Countable Sets of Nodes | 44 |

24 Optimal Nodes | 51 |

25 Functions of Several Variables | 65 |

26 Notes and References | 81 |

Optimal Recovery of Derivatives | 83 |

32 A General Case | 87 |

33 The Periodic Case for the First Derivative | 97 |

34 Noisy Information | 104 |

35 Kolmogorov Inequalities | 113 |

47 Notes and References | 170 |

Quadrature Formulas | 175 |

52 Optimal Quadrature Formulas | 181 |

53 Estimates of Optimal Quadratures | 189 |

54 Notes and References | 198 |

217 | |

219 | |

### Common terms and phrases

27r-periodic functions analytic functions arctan Assume Blaschke products Bojanov bounded linear operator C(TP class Hp classes of analytic complete elliptic integral Consequently consider the problem convex balanced Corollary Denote elliptic functions English transl equality equidistant system exact values exists extremal function extremal problem Fourier coefficients functions analytic Hence Hilbert space Holder inequality holomorphic functions Hp(Bn inequality inf sup infimum information operator intrinsic error Kolmogorov Lemma linear functional lower bound Math method of recovery Micchelli modulus Moreover multi-valued mapping normed linear space optimal method optimal nodes optimal quadrature formula optimal recovery orthonormal basis orthonormal system problem of optimal Proof Proposition proved by Osipenko recovery of functions residue theorem satisfy the condition Section solution Stessin strip Sp subspace sup x',x system of nodes Theorem 1.6 Tikhomirov unit ball unit disk values of n-widths Wilderotter zeros