The paper is a survey of the results obtained in the problems of extremal function interpolation over the past 50 years. Various statements of problems in this direction are analyzed both for the case of one variable and for the case of several variables. A special focus is put on the role of interpolation splines of different types (polynomial, interpolating in the mean, $\cal L$-splines, $m$-harmonic, etc.) in solving the problems of extremal function interpolation. Important applications of the results and methods of extremal interpolation to other problems in approximation theory and the theory of splines are specified.
Keywords: interpolation, splines, approximation, differential operators, difference operators.
The paper was received by the Editorial Office on May 20, 2018.
Yurii Nikolaevich Subbotin, Dr. Phys.-Math. Sci, RAS Corresponding Member, Prof., Krasovskii Institute of Mathematics and Mechanics Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990, Russia, e-mail: yunsub@imm.uran.ru
Sergey Igorevich Novikov, Cand. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990, Russia,
e-mail: Sergey.Novikov@imm.uran.ru
Valerii Trifonovich Shevaldin, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990, Russia,
e-mail: Valerii.Shevaldin@imm.uran.ru
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