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Zhuk, Vladimir Vasilievich

Statistics Math-Net.Ru
Total publications: 50
Scientific articles: 47
Presentations: 2

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This page:2645
Abstract pages:7482
Full texts:2790
References:751
Professor
Doctor of physico-mathematical sciences (1994)
Speciality: 01.01.01 (Real analysis, complex analysis, and functional analysis)
Birth date: 8.05.1940
E-mail:
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.

Biography

Graduated from Faculty of Mathematics and Mechanics of Leningrad State University in 1962 (department of mathematical analysis). D.Sci. thesis was defended in 1994. A list of my works contain more than 140 titles.

   
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.

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List of publications on ZentralBlatt
https://mathscinet.ams.org/mathscinet/MRAuthorID/203934

Publications in Math-Net.Ru
2016
1. M. V. Babushkin, V. V. Zhuk, “On a strong form of asymptotic formulas of Voronovskaya–Bernstein type with pointwise estimate of the remainder term”, Zap. Nauchn. Sem. POMI, 449 (2016),  32–59  mathnet  mathscinet; J. Math. Sci. (N. Y.), 225:6 (2017), 859–876  scopus
2. M. V. Babushkin, V. V. Zhuk, “On two-sided estimates for some functionals in terms of the best approximations”, Zap. Nauchn. Sem. POMI, 449 (2016),  15–31  mathnet  mathscinet; J. Math. Sci. (N. Y.), 225:6 (2017), 848–858  scopus
3. M. V. Babushkin, V. V. Zhuk, “Growth of norms in $L_2$ of derivatives of Steklov functions and properties of functions defined by best approximations and Fourier coefficients”, Zap. Nauchn. Sem. POMI, 445 (2016),  5–32  mathnet  mathscinet; J. Math. Sci. (N. Y.), 222:5 (2017), 525–543  scopus
2015
4. V. V. Zhuk, O. A. Tumka, N. A. Kozlov, “Constants in Jackson-type inequations for the best approximation of periodic differentiable functions”, Vestnik S.-Petersburg Univ. Ser. 10. Prikl. Mat. Inform. Prots. Upr., 2015, 1,  33–41  mathnet  elib
5. V. V. Zhuk, “On strong approximation of functions by positive operators”, Zap. Nauchn. Sem. POMI, 440 (2015),  68–80  mathnet  mathscinet; J. Math. Sci. (N. Y.), 217:1 (2016), 45–53  scopus
2014
6. V. V. Zhuk, O. A. Tumka, “On some modifications of Jackson's generalized theorem for the best approximations of periodic functions”, Vestnik S.-Petersburg Univ. Ser. 10. Prikl. Mat. Inform. Prots. Upr., 2014, 1,  40–50  mathnet
7. V. V. Zhuk, G. Yu. Puerov, “Some inequalities for trigonometric polynomials and Fourier coefficients”, Zap. Nauchn. Sem. POMI, 429 (2014),  64–81  mathnet; J. Math. Sci. (N. Y.), 207:6 (2015), 845–856  scopus
8. V. O. Dron, V. V. Zhuk, “On approximation of periodic functions by modified Steklov averages in $L_2$”, Zap. Nauchn. Sem. POMI, 429 (2014),  20–33  mathnet; J. Math. Sci. (N. Y.), 207:6 (2015), 815–824  scopus
2013
9. O. L. Vinogradov, V. V. Zhuk, “Estimates for functionals with a known finite set of moments in terms of high order moduli of continuity in the spaces of functions defined on the segment”, Algebra i Analiz, 25:3 (2013),  86–120  mathnet  mathscinet  zmath  elib; St. Petersburg Math. J., 25:3 (2014), 421–446  isi  scopus
10. M. V. Babushkin, V. V. Zhuk, “On the constants in inequalities of the generalized Jackson theorem type”, Zap. Nauchn. Sem. POMI, 418 (2013),  28–59  mathnet; J. Math. Sci. (N. Y.), 200:5 (2014), 532–550  scopus
11. O. L. Vinogradov, V. V. Zhuk, “Estimates of functionals by the second moduli of continuity of even derivatives”, Zap. Nauchn. Sem. POMI, 416 (2013),  70–90  mathnet; J. Math. Sci. (N. Y.), 202:4 (2014), 526–540  scopus
2012
12. O. L. Vinogradov, V. V. Zhuk, “Estimates for functional with a known finite set of moments in terms of moduli of continuity and behaviour of constants in the Jackson-type inequalities”, Algebra i Analiz, 24:5 (2012),  1–43  mathnet  mathscinet  zmath  elib; St. Petersburg Math. J., 24:5 (2013), 691–721  isi
13. V. V. Zhuk, “Estimates of best approximations of periodic function by linear combinations values of the function itself and its primitives”, Zap. Nauchn. Sem. POMI, 404 (2012),  157–174  mathnet  mathscinet; J. Math. Sci. (N. Y.), 193:1 (2013), 89–99  scopus
14. V. V. Zhuk, “Inequalities of type generalized Jackson theorem for best approximations”, Zap. Nauchn. Sem. POMI, 404 (2012),  135–156  mathnet  mathscinet; J. Math. Sci. (N. Y.), 193:1 (2013), 75–88  scopus
2011
15. O. L. Vinogradov, V. V. Zhuk, “Estimates for functionals with a known finite set of moments in terms of deviations of operators constructed with the use of the Steklov averages and finite differences”, Zap. Nauchn. Sem. POMI, 392 (2011),  32–66  mathnet; J. Math. Sci. (N. Y.), 184:6 (2012), 679–698  scopus
2010
16. David W. K. Yeung, Leon Petrosyan, Vladimir Zhuk, Anna V. Iljina, “The Detalization of the Irrational Behavior Proof Condition”, Contributions to Game Theory and Management, 3 (2010),  431–440  mathnet
17. O. L. Vinogradov, V. V. Zhuk, “The rate of decrease of constants in Jackson type inequalities in dependence of the order of modulus of continuity”, Zap. Nauchn. Sem. POMI, 383 (2010),  33–52  mathnet
18. O. L. Vinogradov, V. V. Zhuk, “Estimates for functionals with a known moment sequence in terms of deviations of Steklov type means”, Zap. Nauchn. Sem. POMI, 383 (2010),  5–32  mathnet; J. Math. Sci. (N. Y.), 178:2 (2011), 115–131  scopus
2009
19. V. V. Zhuk, “On approximating periodic functions by the Fourier sums”, Zap. Nauchn. Sem. POMI, 371 (2009),  78–108  mathnet; J. Math. Sci. (N. Y.), 166:2 (2010), 167–185  scopus
20. N. Yu. Dodonov, V. V. Zhuk, “On approximating periodic functions by Riesz sums”, Zap. Nauchn. Sem. POMI, 371 (2009),  18–36  mathnet; J. Math. Sci. (N. Y.), 166:2 (2010), 134–144  scopus
2008
21. V. V. Zhuk, “Approximation of periodic functions in the uniform metric by Jackson type polynomials”, Zap. Nauchn. Sem. POMI, 357 (2008),  115–142  mathnet  zmath; J. Math. Sci. (N. Y.), 157:4 (2009), 607–622
22. V. V. Zhuk, “Approximation of periodic functions by Jackson type interpolation sums”, Zap. Nauchn. Sem. POMI, 357 (2008),  90–114  mathnet  zmath; J. Math. Sci. (N. Y.), 157:4 (2009), 592–606
2007
23. V. V. Zhuk, “Approximating periodic functions in Hölder type metrics by the Fourier sums and the Riesz means”, Zap. Nauchn. Sem. POMI, 350 (2007),  70–88  mathnet; J. Math. Sci. (N. Y.), 150:3 (2008), 2045–2055  scopus
2006
24. A. S. Zhuk, V. V. Zhuk, “On approximating periodic functions using linear approximation methods”, Zap. Nauchn. Sem. POMI, 337 (2006),  134–164  mathnet  mathscinet  zmath; J. Math. Sci. (N. Y.), 143:3 (2007), 3090–3107  scopus
25. N. Yu. Dodonov, V. V. Zhuk, “On approximating periodic functions by singular integrals with positive kernels”, Zap. Nauchn. Sem. POMI, 337 (2006),  51–72  mathnet  mathscinet  zmath; J. Math. Sci. (N. Y.), 143:3 (2007), 3039–3052  scopus
2004
26. A. S. Zhuk, V. V. Zhuk, “Some orthogonalities in approximation theory”, Zap. Nauchn. Sem. POMI, 314 (2004),  83–123  mathnet  mathscinet  zmath; J. Math. Sci. (N. Y.), 133:6 (2006), 1652–1675
2002
27. O. L. Vinogradov, V. V. Zhuk, “Sharp Kolmogorov-type inequalities for moduli of continuity and best approximations by trigonometric polynomials and splines”, Zap. Nauchn. Sem. POMI, 290 (2002),  5–26  mathnet  mathscinet  zmath; J. Math. Sci. (N. Y.), 124:2 (2004), 4845–4857
2001
28. V. V. Zhuk, G. I. Natanson, “Semi-norms and continuity modules of functions defined on a segment”, Zap. Nauchn. Sem. POMI, 276 (2001),  155–203  mathnet  mathscinet  zmath; J. Math. Sci. (N. Y.), 118:1 (2003), 4822–4851
1977
29. V. V. Zhuk, “Certain exact bounds for seminorms given on spaces of periodic functions”, Mat. Zametki, 21:6 (1977),  789–798  mathnet  mathscinet  zmath; Math. Notes, 21:6 (1977), 445–450
30. V. V. Zhuk, “Some exact inequalities between the best approximations and moduli of continuity of high orders”, Mat. Zametki, 21:2 (1977),  281–288  mathnet  mathscinet  zmath; Math. Notes, 21:2 (1977), 153–157
1974
31. V. V. Zhuk, “Some sharp inequalities for uniform best approximations of periodic functions”, Dokl. Akad. Nauk SSSR, 214:6 (1974),  1245–1246  mathnet  mathscinet  zmath
1973
32. V. V. Zhuk, “Certain inequalities between best approximations of periodic functions”, Izv. Vyssh. Uchebn. Zaved. Mat., 1973, 9,  18–26  mathnet  mathscinet  zmath
33. V. V. Zhuk, “Certain sharp inequalities between best approximations”, Izv. Vyssh. Uchebn. Zaved. Mat., 1973, 1,  51–56  mathnet  mathscinet  zmath
1972
34. V. V. Zhuk, “The accuracy of the representation of a continuous $2\pi$-periodic function by means of linear approximation methods”, Izv. Vyssh. Uchebn. Zaved. Mat., 1972, 8,  46–59  mathnet  mathscinet  zmath
1971
35. V. V. Zhuk, “Some sharp inequalities between uniform best approximations of periodic functions”, Dokl. Akad. Nauk SSSR, 201:2 (1971),  263–265  mathnet  mathscinet  zmath
36. V. V. Zhuk, “Some exact inequalities between best approximations and moduli of continuity”, Dokl. Akad. Nauk SSSR, 196:4 (1971),  748–750  mathnet  mathscinet  zmath
1970
37. V. V. Zhuk, “The rate of approximation of a continuous $2\pi$-periodic function by partial sums of its Fourier series”, Dokl. Akad. Nauk SSSR, 190:5 (1970),  1015–1018  mathnet  mathscinet  zmath
38. V. V. Zhuk, “Some relations between moduli of continuity and functionals defined on sets of periodic functions”, Izv. Vyssh. Uchebn. Zaved. Mat., 1970, 5,  24–33  mathnet  mathscinet  zmath
1969
39. V. V. Zhuk, “The order of approximation of a continuous $2\pi$-periodic function by linear methods”, Izv. Vyssh. Uchebn. Zaved. Mat., 1969, 10,  40–50  mathnet  mathscinet  zmath
40. V. V. Zhuk, G. I. Natanson, “Saturation theory converse problem”, Mat. Zametki, 6:5 (1969),  583–590  mathnet  mathscinet  zmath; Math. Notes, 6:5 (1969), 811–815
1968
41. V. V. Zhuk, “The approximation of periodic functions by linear approximation methods”, Dokl. Akad. Nauk SSSR, 179:5 (1968),  1038–1041  mathnet  mathscinet  zmath
42. V. V. Zhuk, “On the order of approximation of a continuous $2\pi$-periodic function by Fejer and Poisson means of its Fourier series”, Mat. Zametki, 4:1 (1968),  21–32  mathnet  mathscinet  zmath; Math. Notes, 4:1 (1968), 500–508
1967
43. V. V. Zhuk, “Approximation of periodic functions by linear methods of summation of Fourier series”, Dokl. Akad. Nauk SSSR, 173:1 (1967),  30–33  mathnet  mathscinet  zmath
1966
44. V. V. Zhuk, “Approximation of periodic functions bounded by a subadditive operator”, Dokl. Akad. Nauk SSSR, 169:3 (1966),  515–518  mathnet  mathscinet  zmath
1965
45. V. V. Zhuk, “Some modifications of the concept of modulus of smoothness and their applications”, Dokl. Akad. Nauk SSSR, 162:1 (1965),  19–22  mathnet  mathscinet  zmath
46. V. V. Zhuk, “A modification of the concept of modulus of smoothness and its application to the estimation of Fourier coefficients”, Dokl. Akad. Nauk SSSR, 160:4 (1965),  758–761  mathnet  mathscinet  zmath
47. V. V. Zhuk, “On the absolute convergence of Fourier series”, Dokl. Akad. Nauk SSSR, 160:3 (1965),  519–522  mathnet  mathscinet  zmath

2004
48. V. M. Babich, A. M. Vershik, V. S. Videnskii, O. L. Vinogradov, I. K. Daugavet, N. Yu. Dodonov, V. V. Zhuk, B. M. Makarov, A. N. Podkorutov, Yu. G. Reshetnyak, M. A. Skopina, V. L. Fainshmidt, V. P. Havin, N. A. Shirokov, “Garal'd Isidorovich Natanson (obituary)”, Uspekhi Mat. Nauk, 59:4(358) (2004),  181–185  mathnet  mathscinet  zmath; Russian Math. Surveys, 59:4 (2004), 771–776  isi
2002
49. V. V. Zhuk, V. N. Malozemov, G. I. Natanson, V. P. Havin, “Viktor Solomonovich Videnskii (on his 80th birthday)”, Uspekhi Mat. Nauk, 57:5(347) (2002),  182–186  mathnet  mathscinet  zmath; Russian Math. Surveys, 57:5 (2002), 1033–1038  isi
2001
50. E. G. Goluzina, V. V. Zhuk, G. V. Kuz'mina, N. A. Shirokov, “Nikolai Andreevich Lebedev and the Leningrad school of function theory in the 1950–1970s”, Zap. Nauchn. Sem. POMI, 276 (2001),  5–19  mathnet  mathscinet  zmath; J. Math. Sci. (N. Y.), 118:1 (2003), 4733–4739

Presentations in Math-Net.Ru
1. Inequalities on best approximations of periodic functions like generalized Jackson theorem
V. V. Zhuk
International conference "Nonlinear Approximations and Applications" dedicated to the 60th birthday of Professor V. N. Temlyakov
October 31, 2013 16:40   
2. Best approximations and functions of the class $C^r$
V. V. Zhuk
Meetings of the St. Petersburg Mathematical Society
April 15, 1997

Organisations
 
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