Speciality:
01.01.01 (Real analysis, complex analysis, and functional analysis)
Birth date:
8.05.1940
Keywords: precision of approximation of functions; trigonometrical Fourier series; restoration of functions by values in points; series summation; extremal problems of approximation theory; linear methods of approximation; inequalities for derivatives; formulas of numerical differentiation; moduli of continuity; strong approximation of functions.
Subject:
Scientific interests are connected with approximation theory, Fourier series and their applications. General theorems which allow to get both-sided estimates for deviation of the wide class of approximation methods in the terms of moduli of continuity are established. These estimates coincide if one doesn"t take constants into account and are sharp in the order sense for each individual function. The techniqie of obtaining the estimates of approximation methods by moduli of continuity of arbitrary order of functions defined on the line or on the segment is developed. The constants in these estimates are greatly more sharp in comparison with the constants which were known earlier. Several difficult extremal problems are solved. These problems deal with finding of sharp constants in direct theorems of approximation theory (Jackson-type inequalities) and inequalities for derivatives (Landau–Kolmogorov-type inequalities). These problems were studied in connection with each other for the first time. Some "latent" orthogonalities connecting important for approximation theory objects are discovered. The analogs of Parseval equality are established and their applications to different problems, especially to the strong approximation, are given. Strictly mathematically justified, simple and effective algorithms of rectoration of function of several variables by its values in given points are constructed. New results concerning convergence of ordinary and multiple Fourier series are obtained. In 1999&ndash2001 the series of papers (jointly with O. L. Vinogradov) was published. These papers deal with extremal problems of approximation theory which lend themselves to solving very slowly.
Main publications:
Zhuk V. V. Approksimatsiya periodicheskikh funktsii. Leningrad, 1982. 366 s.
Zhuk V. V. Silnaya approksimatsiya periodicheskikh funktsii. Leningrad, 1989. 296 s.
Zhuk V. V., Kuzyutin V. F. Approksimatsiya funktsii i chislennoe integrirovanie. S.-Peterburg, 1995. 352 s.
Vinogradov O. L., Zhuk V. V. Tochnye otsenki pogreshnostei formul tipa chislennogo differentsirovaniya na trigonometricheskikh mnogochlenakh // Problemy matematicheskogo analiza. Vypusk 21. 2000. S. 68–109.
Vinogradov O. L., Zhuk V. V. Tochnye neravenstva tipa Dzheksona dlya differentsiruemykh funktsii i minimizatsiya shaga modulya nepreryvnosti // Trudy S.-Peterburgskogo matematicheskogo obschestva. T. 8. 2000. S. 29–51.
