A Class of I.M. Vinogradov’s Series and Its Applications in Harmonic Analysis

  • K. I. Oskolkov
Conference paper
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 19)

Abstract

The present paper is a survey of the author’s recent research in the one-dimensional trigonometric series of the type
$$\sum\limits_n {\hat f\left( n \right)} {e^{2\pi i\left( {{n^r}{x_r} + \cdots + n{x_1}} \right)}}.$$
(1.1)

Keywords

Manifold Convolution Hunt Stein E211 

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References

  1. [1]
    G.I. Arkhipov, On the Hilbert-Kamke Problem, Izv. Akad. Nauk SSSR, Ser Mat., 48(1984), 3–52; English transl. in Math. USSR Izv., 24(1985).MathSciNetMATHGoogle Scholar
  2. [2]
    G.I. Arkhipov and K.I. Oskolkov, On a special trigonometric series and its applications, Matem Sbornik, 134(176) (1987), N2; Engl, transl. in Math. USSR Sbornik, 62(1989), N1, 145–155.Google Scholar
  3. [3]
    N.K. Bari, A Treatise on Trigonometric Series, v. 1, (Pergamon Press, N.Y.), 1964.Google Scholar
  4. [4]
    L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math., 116(1966), 133–157.MathSciNetCrossRefGoogle Scholar
  5. [5]
    Yung-Ming Chen, A remarkable divergent Fourier series, Proc. Japan Acad., 38(1962), 239–244.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    L. de Michele and P.M. Soardi, Uniform convergence of lacunary Fourier series, Colloq. Math., 36(1976), 285–287.MathSciNetMATHGoogle Scholar
  7. [7]
    H. Fiedler, W. Jurkat, and O. Koerner, Asymptotic expansion of finite theta series, Acta Arithmetica, 32(1977), 129–146.MathSciNetMATHGoogle Scholar
  8. [8]
    A. Figa-Talamanca, An example in the theory of lacunary Fourier series, Boll. Un. Mat. Ital. (4), 3(1970), 375–378.MathSciNetMATHGoogle Scholar
  9. [9]
    J.E. Fournier and L. Pigno, Analytic and arithmetic properties of thin sets, Pacific J. Math., 105(1983), 115–141.MathSciNetMATHGoogle Scholar
  10. [10]
    R.P. Gosselin, On the divergence of Fourier series, Proc. Amer. Math. Soc, 9(1958), 278–282.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    G.H. Hardy, Collected Papers of G.H. Hardy (Oxford: Clarendon Press), 1966, v. 1.MATHGoogle Scholar
  12. [12]
    G.H. Hardy and J.E. Littlewood, Some problems of Diophantine approximation. II. The trigonometrical series associated with the elliptic θ-functions, Acta Math., 37(1914), 193–238.MathSciNetCrossRefGoogle Scholar
  13. [13]
    R.A. Hunt, An estimate of the conjugate function, Studia Math., 44(1972), 371–377.MathSciNetGoogle Scholar
  14. [14]
    R.A. Hunt, On the convergence of Fourier series. Orthogonal Expansions and Their Continuous Analogues. (Proc Conf. Edwardsville, Ill. (1967)), 235–255. Southern Ill. Univ. Press, Carbondale, Ill., (1968).Google Scholar
  15. [15]
    Chen Jing-run, On Professor Hua’s estimate of exponential sums, Sci. Sinica, 20 (1977), 711–719.MathSciNetMATHGoogle Scholar
  16. [16]
    A.N. Kolmogorov, Sur les fonctions harmoniques conjugees et les series de Fourier, Fund. Math., 7(1925), 24–29.Google Scholar
  17. [17]
    A.N. Kolmogorov, Une serie de Fourier Lebesgue divergente partout, CR. Acad. Sei Paris, 183(1926), 1327–1328.Google Scholar
  18. [18]
    A.N. Kolmogorov, Une série de Fourier-Lebesgue divergente presque partout, Fund. Math., 4(1923), 324–328.Google Scholar
  19. [19]
    S.V. Konyagin, On Littlewood’s conjecture, Izv. Akad. Nauk SSSR Ser. Mat., 45(1981), 243–265.MathSciNetMATHGoogle Scholar
  20. [20]
    E. Makai, On the summability of the Fourier series of L 2 ü2. IV. Acta Math. Ac. Sei Hung., 20(1969), 383–391.MathSciNetMATHCrossRefGoogle Scholar
  21. [21]
    K.I. Oskolkov, I.M. Vinogradov series and integrals and their applications, Trudy Mat. Inst Steklov, 190(1989), 186–221.MathSciNetMATHGoogle Scholar
  22. [22]
    K.I. Oskolkov, I.M. Vinogradov’s series in the Cauchy problem for Schroedinger type equations, Trudy Mat. Inst. Steklov, 200(1991) (in print).Google Scholar
  23. [23]
    K.I. Oskolkov, On functional properties of incomplete Gaussian sums, Canad. J. Math., 43(1991), No. 1, 182–212.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    K.I. Oskolkov, On properties of a class of Vinogradov series, Doklady Acad. Nauk SSSR, 300(1988), N4, 737–741; Engl, transl. in Soviet Math. Dokl., 37(1988), N3.MathSciNetGoogle Scholar
  25. [25]
    K.I. Oskolkov, On spectra of uniform convergence, Dokl. Akad. Nauk SSSR, 288(1986), N1; Engl, transl. in Soviet Math. Dokl, 33(1986), N3, 616–620.MathSciNetGoogle Scholar
  26. [26]
    K.I. Oskolkov, Subsequences of Fourier sums of integrable functions, Trudy Mat. Inst. Steklov, 167(1985), 239–360; Engl, transl. in Proc. Steklov Inst. Math. 1986, N2 (167).MathSciNetMATHGoogle Scholar
  27. [27]
    L. Pedemonte, Sets of uniform convergence, Colloq. Math., 33(1975), 123–132.MathSciNetMATHGoogle Scholar
  28. [28]
    N. Saitô and Y. Aizawa, editors, Progress of Theoretical Physics, Supplement, No. 98,1989. New trends in chaotic dynamics of Hamiltonian systems, Kyoto University, Japan.Google Scholar
  29. [29]
    B. Smith, O.C. McGehee, L. Pigo, Hardy’s inequality and L 1-norm of exponential sums, Ann. Math. 113(1981), N3, 613–618.MATHCrossRefGoogle Scholar
  30. [30]
    S.B. Stechkin, Estimate of a complete rational trigonometric sum, Trudy Mat. Inst. Steklov, 143(1977), 188–207; English transl. in Proc. Steklov Inst. Math. 1980, N1 (143).MathSciNetGoogle Scholar
  31. [31]
    S.B. Stechkin, On absolute convergence of Fourier series. III, Izv. Akad. Nauk SSSR Ser. Mat., 20(1956), 385–412. (Russian).Google Scholar
  32. [32]
    E.M. Stein, On limits of sequences of operators, Ann. Math., 74(1961), 140–170.MATHCrossRefGoogle Scholar
  33. [33]
    E.M. Stein, Oscillatory integrals in Fourier analysis, In: Beijing Lectures in Harmonic Analysis, Princeton Univ. Press, Princeton, N.J., (1986), 307–355.Google Scholar
  34. [34]
    E.M. Stein and S. Wainger, The estimation of an integral arising in multiplier transformations, Studia Math., 35(1970), 101–104.MathSciNetMATHGoogle Scholar
  35. [35]
    E.M. Stein and S. Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc, 84(1978), 1239–1295.MathSciNetMATHCrossRefGoogle Scholar
  36. [36]
    V. Totik, On the divergence of Fourier series, Publ. Math. Debrecen, 29(1982), 251–264.MathSciNetMATHGoogle Scholar
  37. [37]
    G. Travaglini, Some properties of UC-sets, Boll. Un. Math. Ital. B(5), v. 15 (1978), 275–284.MathSciNetGoogle Scholar
  38. [38]
    P.L. Ul’yanov, Some questions in the theory of orthogonal and biorthogonal series, Izv. Akad. Nauk Azerbaidzhan. SSR Ser. Fiz.-Tekhn. Math. Nauk (1965), no. 6, 11–13 (Russian).Google Scholar
  39. [39]
    I.M. Vinogradov, The Method of Trigonometric Sums in Number Theory, 2nd ed. “Nauka,” Moscow 1980; English transl. in his Selected Works, Springer Verlag, 1985.MATHGoogle Scholar
  40. [40]
    S. Wainger, Applications of Fourier transforms to averages over lower dimensional sets, Proc. Symposia in Pure Math., 35 (part 1), 1979, 85–94.Google Scholar
  41. [41]
    S. Wainger, Averages and singular integrals over lower dimensional sets, In: Beijing Lectures in Harmonic Analysis, Princeton Univ. Press, Princeton, N.J., (1986), 357–421.Google Scholar
  42. [42]
    S. Wainger, On certain aspects of differentiation theory, Topics in modern harmonic analysis (Proc. Seminar Torino and Milano, May-June 1982), 42(Roma, 1983), 667–706.Google Scholar
  43. [43]
    D.M. Wardlaw and W. Jaworski, Time delay, resonances, Riemann zeros and Chaos in a model quatum scattering system, J. Phys. A: Math. Gen., 22(1989), 3561–3575.MathSciNetMATHCrossRefGoogle Scholar
  44. [44]
    H. Weyl, Über die Gleichverteilung der Zahlen mod Eins, Math. Ann., 77(1915/16), 313–352.MathSciNetGoogle Scholar
  45. [45]
    A. Zygmund, Trigonometric Series, 2nd rev. ed., v. 1, Cambridge Univ. Press. 1959.MATHGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1992

Authors and Affiliations

  • K. I. Oskolkov
    • 1
  1. 1.Department of Math. & Statistics Jeffrey HallQueen’s UniversityKingstonCanada

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