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Theory of interpolation in problem form. (Теория интерполирования в задачах.) (Russian) Zbl 1236.41005
Minsk: Izdat. “Vyšèĭšaja Škola”. 318 p. (1968).
Summary: One third of this book is given to the statement of problems, whose solutions appear in the remaining two thirds.
The first chapter offers 101 problems, most of them concerned with the construction of yet another form for the interpolating algebraic or trigonometric polynomial suitable for yet another particular set-up. In preparation for the remaining chapters, trigonometric polynomials are heavily stressed, while such practically important concepts as divided differences receive scant attention.
Chapter 2, with just 16 problems, acquaints the student with the Lozinskiĭ-Haršiladze theorem and Faber’s theorem, and other evidence of divergence of sequences of interpolating polynomials, including the construction of a continuous periodic function for which the sequence of trigonometric polynomials interpolating at equispaced points diverges everywhere.
The next three chapters investigate how convergence can be achieved by the customary means of restricting the interpoland or modifying the sequence of interpolating polynomials. Specifically, Chapter 3, with 72 problems, develops sharp bounds for certain polynomial operators (i.e., linear projectors with polynomial range), and then considers the convergence of Čebyšev series on \([-1,1]\) for functions analytic on an ellipse, the convergence (uniformly and mean square) of the sequence of polynomials interpolating at the zeros of orthogonal polynomials, the analog of Fejér’s theorem for trigonometric polynomials, and the convergence of a sequence of polynomials interpolating a differentiable function. Chapter 4 concerns generalizations of the Fejér sum, while Chapter 5 investigates in detail the convergence behavior of a sequence of interpolating polynomials in case the interpoland satisfies a Lipschitz condition.
The 58 problems in Chapter 6 center around the question of “best”. interpolation points in \([-1,1]\), i.e., of interpolation points which minimize a suitable norm of the polynomial operator of interpolation at a fixed number of points. The final chapter deals with the special case of the Hermite-Birkhoff problem when agreement in value and second derivative at a number of points is asked for.
A student without a good grasp of classical analysis, or a course in complex analysis, or an introduction to functional analysis, or a love for formula manipulation will find this book very hard going indeed. Beyond the choice of problems and their order (and, of course, their solution), the author gives no guidance to the student, so that this book does not seem suitable for self-study. But as a collection of often very interesting and certainly challenging problems for the student taking a course in constructive function theory, this book is to be highly recommended.
For Part 2 see Zbl 1236.41006.

41A05 Interpolation in approximation theory
41-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to approximations and expansions
00A07 Problem books