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On the Scientific Work of Konstantin Ilyich Oskolkov

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Recent Advances in Harmonic Analysis and Applications

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Abstract

This chapter is a brief account of the life and the scientific work of K.I. Oskolkov.

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References

Scientific Articles by K.I. Oskolkov

  1. Andreev, A., Berdyshev, V.I., Bojanov, B., Kashin, B.S., Konyagin, S.V., Nikol’skii, S.M., Oskolkov, K.I., Petrushev, P., Sendov, B., Telyakovskii, S.A., Temlyakov, V.N.: In memory of Sergei Borisovich Stechkin [1920–1995], East J. Approx. 2, 131–133 (1996).

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  2. Ciesielski, Z. Approximation by algebraic polynomials on simplexes. (Russian) Translated from the English by K. I. Oskolkov. Uspekhi Mat. Nauk, 4(244), 212–214 (1985).

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  3. DeVore, R.A., Oskolkov, K.I., Petrushev, P.P.: Approximation by feed-forward neural networks, Ann. Numer. Math. 4, 261–287 (1997). The heritage of P.L. Chebyshev: A Festschrift in honor of the 70th birthday of T.J. Rivlin. MR 97i:41043

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  4. DeVore, R.A., Oskolkov, K.I., Petrushev, P.P.: Approximation by feed-forward neural networks. [J] Ann. Numer. Math. 4(1–4), 261–287 (1997). ISSN 1021–2655

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  5. Maiorov, V.E., Oskolkov, K., Temlyakov, V.N.: In: Bojanov, B. (ed.) Gridge approximation and Radon compass, Approximation Theory: a Volume-Dedicated to Blagovest Sendov, pp. 284–309, DARBA, Sofia (2002)

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  6. Offin, D., Oskolkov, K.I.: A note on orthonormal polynomial bases and wavelets. Constr. Approx. 9, 319–325 (1993). MR 94f:42047

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  7. Oskolkov, K.I.: Convergence of a trigonometric series to a function of bounded variation. Mat. Zametki 8, 47–58 (1970) (Russian)

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  8. Oskolkov, K.I., Steckin, S.B., Teljakovskii, S.A.: Petr Vasil’evic Galkin, Mat. Zametki 10, 597–600 (1971) MR 44 No 6436 (Russian)

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  9. Oskolkov, K.I.: The norm of a certain polynomial operator. Sibirsk Mat. Z. 12, 1151–1157 (1971) MR 45 No 4021 (Russian)

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  10. Oskolkov, K.I., Teljakovskii, S.A.: On the estimates of P. L. Ul’janov for integral moduli of continuity. Izv. Akad. Nauk Armjan. SSR Ser. Mat. 6, 406–411 (1971). MR 45 No 8782 (Russian)

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  11. Oskolkov, K.I.: The sharpness of the Lebesgue estimate for the approximation of functions with prescribed modulus of continuity by Fourier sumes, Trudy Mat. Inst. Steklov. 112, 337–345 (1971), 389, Collection of articles dedicated to Academician Ivan Matveevic Vinogradov on his 80th birthday, I. MR 49 No No 970 (Russian)

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  12. Oskolkov, K.I.: Generalized variation, the Banach indicatrix and the uniform convergence of Fourier series, Mat. Zametki 12, 313–324 (1972). MR 47 No 5507 (Russian)

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  13. Oskolkov, K.I.: Subsequences of Fourier sums of functions with a prescribed modulus of continuity. Mat. Sb. (N.S.) 88(130), 447–469 (1972). MR 48 No 11874 (Russian)

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  14. Oskolkov, K.I.: Fourier sums for the Banach indicatrix. Mat. Zametki 15, 527–532 (1974). MR 50 No 10177 (Russian)

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  15. Oskolkov, K.I.: Estimation of the rate of approximation of a continuous function and its conjugate by Fourier sums on a set of full measure. Izv. Akad. Nauk SSSR Ser. Mat. 38, 1393–1407 (1974). MR 50 No 10663 (Russian)

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  16. Oskolkov, K.I.: An estimate for the approximation of continuous functions by sequences of Fourier sums, Trudy Mat. Inst. Steklov. 134, 240–253 (1975), 410, Theory of functions and its applications (collection of articles dedicated to Sergei Mikhailovich Nikolskii on the occasion of his 70th birthday). MR 53 No 6203 (Russian)

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  17. Oskolkov, K.I.: Lebesgue’s inequality in the uniform metric and on a set of full measure. Mat. Zametki 18, 515–526 (1975). MR 54 No 833 (Russian)

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  18. Oskolkov, K.I.: On strong summability of Fourier series and differentiability of functions. Anal. Math. 2, 41–47 (1976). MR 53 No 6210 (English, with Russian summary)

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  19. Oskolkov, K.I.: The uniform modulus of continuity of summable functions on sets of positive measure. Dokl. Akad. Nauk SSSR 229, 304–306 (1976). MR 57 No 9917 (Russian)

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  20. Oskolkov, K.I.: Approximation properties of integrable functions on sets of full measure. Mat. Sb. (N.S.) 103(145), 563–589 (1977), 631. MR 57 No 13343 (Russian)

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  21. Oskolkov, K.I.: Sequences of norms of Fourier sums of bounded functions, Trudy Mat. Inst. Steklov. 142, 129–142 (1977), 210, Analytic number theory, mathematical analysis and their applications (dedicated to I.M. Vinogradov on his 85th birthday).

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  22. Oskolkov, K.I.: Polygonal approximation of functions of two variables, Mat. Sb. (N.S.) 107(149), 601–612 (1978), 639. MR 81j:41020 (Russian)

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  23. Oskolkov, K.I.: Quantitative estimates of N.N. Luzin’s C-property for classes of integrable functions, Approximation Theory (Papers, VIth Semester, Stefan Banach Internat. Math. Center, Warsaw, 1975), Banach Center Publ. 4 PWN, Warsaw (1979), 185–196 MR 81a:26003

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  24. Oskolkov, K.I.: Optimality of a quadrature formula with equidistant nodes on classes of periodic functions. Dokl. Akad. Nauk SSSR 249, 49–52 (1979). MR 81b:41077 (Russian)

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  25. Oskolkov, K.I.: Lebesgue’s inequality in the mean. Mat. Zametki 25, 551–555 (1979), 636. MR 81c:42005 (Russian)

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  26. Oskolkov, K.I.: The upper bound of the norms of orthogonal projections onto subspaces of polygonals, Approximation Theory (Papers, VIth Semester, Sefan Banach Internat. Math. Center, Warsaw, 1975), Banach Center Publ., 4, PWN, Warsaw (1979), 177–183. MR 82e:41013

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  27. Oskolkov, K.I.: Approximate properties of classes of periodic functions. Mat. Zametki 27, 651–666 (1980). MR 81j:42011 (Russian)

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  28. Oskolkov, K.I.: Partial sums of the Taylor series of a bounded analytic function. Trudy Mat. Inst. Steklov. 157, 153–160 (1981), 236, Number Theory, mathematical analysis and their applications. MR 83c:300004 (Russian)

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  29. Oskolkov, K.I.: On optimal quadrature formulas on certain classes of periodic functions. Appl. Math. Optim. 8, 245–263 (1982). MR 83h:41032

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  30. Oskolkov, K.I.: On exponential polynomials of the least Lp-norm, Constructive Function Theory ’81 (Varna, 1981), Publ. House Bulgar. Acad. Sci., Sofia (1983), 464–467. MR 85a:41022

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  31. Oskolkov, K.I.: Luzin’s C-property for a conjugate function. Trudy Mat. Inst. Steklov. 164, 124–135 (1983). Orthogonal series and approximations of functions. MR 86e:42019 (Russian)

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  32. Oskolkov, K.I.: Strong summability of Fourier series Trudy Mat. Inst. Steklov. 172, 280–290, 355 (1985) Studies in the theory of functions of several real variables and the approximation of functions. MR 87a:42021 (Russian)

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  33. Oskolkov, K.I.: A subsequence of Fourier sums of integrable functions. Trudy Mat. Inst. Steklov 167, 239–260, 278 (1985) Current problems in mathematics. Mathematical analysis, algebra, topology. MR 87i:42008 (Russian)

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  34. Oskolkov, K.I.: Spectra of uniform convergence. Dokl. Akad. Nauk. SSSR 288, 54–58 (1986). MR 88e:42012 (Russian)

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  35. Oskolkov, K.I.: Inequalities of the “large size” type and applicatiojns to problems of trigonometric approximation. Anal. Math. 12, 143–166 (1986). MR 88i:42002 (English, with Russian summary)

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  36. Arkhipov, G.I., Oskolkov, K.I.: A special trigonometric series and its applications. Mat. Sb. (N.S.) 134(176), 147–157, 287 (1987). MR 89a:42010 (Russian)

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  37. Oskolkov, K.I.: Continuous functions with polynomial spectra, Investigations in the theory of the approximation of functions (Russian). Akad. Nauk SSSR Bashkir. Filial Otdel Fiz. Mat., Ufa,, 187–200 (1987). MR 90b:42013 (Russian)

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  38. Oskolkov, K.I.: Properties of a class of I.M. Vinogradov series. Dokl. Akad. Nauk SSSR 300, 803–807 (1988). MR 89f:11117 (Russian)

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  39. Oskolkov, K.I.: I.M. Vinogradov series and integrals and their applications, Trudy Mat. Inst. Steklov. 190, 186–221 (1989), Translated in Proc. Steklov Math. 1992, no. 1, 193–229; Theory of functions (Russian) (Amberd, 1987). MR 90g:11112 (Russian)

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  40. Oskolkov, K.I.: On functional properties of incomplete Gaussian sums. Canad. J. Math. 43, 182–212 (1991). MR 92e:11083

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  41. Oskolkov, K.I.: I.M. Vinogradov series in the Cauchy problem for Schrödinger-type equations. Trudy Mat. Inst. Steklov. 200, 265–288 (1991). MR 93b:11104 (Russian)

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  42. Oskolkov, K.I.: A class of I.M. Vinogradov’s series and its applications in harmonic analysis, Progress in Approximation Theory (Tampa, FL, 1990), Springer Ser. Comput. Math. 19, 353–402. Springer, New York (1992). MR 94m:42016

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  43. Oskolkov, K.I.: Ridge approximation, Fourier-Chebyshev analysis, and optimal quadrature formulas. Tr. Mat. Inst. Steklov 219, 269–285 (1997). MR 99j:41036 (Russian)

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  44. Oskolkov, K.I.: Schrödinger equation and oscillatory Hilbert transforms of second degree. J. Fourier Anal. Appl. 4, 341–356 (1998). MR 99j:42004

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  45. Oskolkov, K.I.: Ridge approximation and the Kolmogorov-Nikolskii problem. Dokl. Akad. Nauk 368, 445–448 (1999). MR 2001b:41024 (Russian)

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  46. Oskolkov, K.I.: Linear and nonlinear methods for ridge approximation, Metric theory of functions and related problems in analysis (Russian), Izd. Nauchno-Issled. Aktuarno-Finans. Tsentra (AFTs), Moscow, 165–195 (1999). MR 2001i:41039 (Russian, with Russian summary)

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  47. Oskolkov, K.I.: Ridge approximations and the Kolmogorov-Nikol’skij problem. [J] Dokl. Math. 60(2), 209–212 (1999); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 368(4), 445–448 (1999). ISSN 1064–5624; ISSN 1531–8362

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  48. Oskolkov, K.I.: On representations of algebraic polynomials by superpositions of plane waves. [J] Serdica Math. J. 28(4), 379–390 (2002). ISSN 0204–4110

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  49. Maiorov, V.E., Oskolkov, K.I., Temlyakov, V.N.: Gridge approximation and Radon compass. [A] Bojanov, B.D. (ed.) Approximation theory. A volume dedicated to Blagovest Sendov. Sofia: DARBA. 284–309 (2002). ISBN 954-90126-5-4/hbk

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  50. Oskolkov, K.: On representations of algebraic polynomials by superpositions of plane waves. Serdica Math. J. 28, 379–390 (2002). Dedicated to the memory of Vassil Popov on the occasion of his 60th birthday

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  51. Oskolkov, K.I.: Continued fractions and the convergence of a double trigonometric series. [J] East J. Approx. 9(3), 375–383 (2003). ISSN 1310–6236

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  52. Oskolkov, K.: Continued fractions and the convergence of a double trigonometric series. East J. Approx. 9, 375–383 (2003)

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  53. Oskolkov, K.: On a result of Telyakovskii and multiple Hilbert transforms with polynomial phases. Mat. Zametki 74, 242–256 (2003)

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  54. Oskolkov, K.: Schrödinger equation and oscillatory Hilbert transforms of second degree. J. Fourier Anal. Appl. 4, 341–356 (1998)

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  55. Oskolkov, K.I.: The series \(\sum \nolimits \frac{{e}^{2\pi inmx}} {mn}\) and Chowla’s problem. Proc. Steklov Inst. Math. 248, 197–215 (2005)

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  56. Oskolkov, K.I.: The Schrödinger density and the Talbot effect. [A] Figiel, T. (ed.) et al., Approximation and probability. Papers of the conference held on the occasion of the 70th anniversary of Prof. Zbigniew Ciesielski, Bedlewo, Poland, September 20–24, 2004. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 72, 189–219 (2006).

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  57. Oskolkov, K.I.: Linear and nonlinear methods of relief approximation. (English. Russian original) J. Math. Sci., New York 155(1), 129–152 (2008); translation from Sovrem. Mat., Fundam. Napravl. 25, 126–148 (2007). ISSN 1072–3374; ISSN 1573–8795

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  58. Oskolkov, K.I.; Chakhkiev, M.A.: On Riemann “nondifferentiable” function and Schrödinger equation. Proc. Steklov Inst. Math. 269, 186–196 (2010); translation from Trudy Mat. Inst. Steklova 269, 193–203 (2010).

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  59. Tandori, K. Systems of signs. (Russian) Translated from the German by K. I. Oskolkov. International conference on current problems in algebra and analysis (Moscow-Leningrad, 1984). Uspekhi Mat. Nauk, 4(244), 105–108 (1985).

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Books and Articles Translated or Edited by K.I.Oskolkov

  1. Brensted, A.: Vvedenie v teoriyu vypuklykh mnogogrannikov, “Mir”, Moscow, 1988,Translated from the English by K.I. Oskolkov; Translation edited and with a preface by B.S. Kashin. (Russian)

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  2. Sendov, B., Popov, V.: Usrednennye moduli gladkosti, “Mir”, Moscow, 1988, Translated from the Bulgarian and with a preface by Yu. A. Kuznetsov and K.I. Oskolkov. (Russian)

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  3. Vinogradov, I.M., Karacuba, A.A., Oskolkov, K.I., Parsin, A.N.: Trudy mezhdunarodnoi konferenctsii po teorii chisel (Moskva, 14–18 sentyabrya 1971 g.), Izdat. “Nauka”, Moscow, 1973, With an introductory address by M.V. Keldys; Trudy Mat. Inst. Steklov. 132 (1973). (Russian)

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Authors and Affiliations

  1. University of South Carolina, Columbia, SC, 29208, USA

    Dmitriy Bilyk

  2. Florida International University, Miami, FL, 33199, USA

    Laura De Carli

  3. Georgia Southern University, Statesboro, GA, 30460, USA

    Alexander M. Stokolos

  4. University of Georgia, Athens, GA, 30602, USA

    Alexander Petukhov

  5. Georgia Institute of Technology, Atlanta, GA, 30308–0332, USA

    Brett D. Wick

Authors
  1. Dmitriy Bilyk
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  2. Laura De Carli
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  3. Alexander Petukhov
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Correspondence to Alexander M. Stokolos .

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Editors and Affiliations

  1. , Department of Mathematics, University of South Carolina, Greene Street 1523, Columbia, 29208, South Carolina, USA

    Dmitriy Bilyk

  2. University Park, Department of Mathematics, Florida International University, SW 8th St., Rm 416 11200, Miami, 33199, Florida, USA

    Laura De Carli

  3. , Department of Mathematics, Universtity of Georgia, University of Georgia, Athens, 30602, Georgia, USA

    Alexander Petukhov

  4. Georgia Southern University, PO Box 8093, Statesboro, 30460, Georgia, USA

    Alexander M. Stokolos

  5. , Department of Mathematics, Georgia Institute of Technology, Cherry Street 686, Atlanta, 30332, Georgia, USA

    Brett D. Wick

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Bilyk, D., De Carli, L., Petukhov, A., Stokolos, A.M., Wick, B.D. (2012). On the Scientific Work of Konstantin Ilyich Oskolkov. In: Bilyk, D., De Carli, L., Petukhov, A., Stokolos, A., Wick, B. (eds) Recent Advances in Harmonic Analysis and Applications. Springer Proceedings in Mathematics & Statistics, vol 25. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4565-4_1

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