SOME EXTREMAL PROBLEMS OF HARMONIC ANALYSIS AND APPROXIMATION THEORY

The paper is devoted to a survey of the main results obtained in the solution of the Tura´n and Fejer extremal problems on the torus; the Tur´an, Delsarte, Bohmann, and Logan extremal problems on the Euclidean space, half-line, and hyperboloid. We also give results obtained when solving a similar problem on the optimal argument in the module of continuity in the sharp Jackson inequality in the space L² on the Euclidean space and half-line. Most of the results were obtained by the authors of the review. The survey is based on a talk made by V.I. Ivanov at the conference «6th Workshop on Fourier Analysis and Related Fields, Pecs, Hungary, 2431 August 2017». We solve also the problem of the optimal argument on the hyperboloid. As the basic apparatus for solving extremal problems on the half-line, we use the Gauss and Markov quadrature formulae on the half-line with respect to the zeros of the eigenfunctions of the Sturm–Liouville problem. For multidimensional extremal problems we apply a reduction to one-dimensional problems by means of averaging of admissible functions over the Euclidean sphere. Extremal function is unique in all cases.

Fourier, Hankel, and Jacobi transforms, Tura´n, Fej´er, Delsarte, Bohman, and Logan extremal problems, Gauss and Markov quadrature formulae

1. Stechkin S.B., 1972, “An extremal problem for trigonometric series with nonnegative coefficients”, Acta Math. Acad. Sci. Hung., vol. 23, № 3-4, pp. 289–291.

2. Gorbachev D.V., Manoshina A.S., 2004, “Tur´an Extremal Problem for Periodic Functions with Small Support and Its Applications”, Math. Notes, vol. 76, № 5, pp. 640–652.

3. Fej´er L., 1915, “¨Uber trigonometrische Polynome”, J. Angew. Math., vol. 146, pp. 53–82.

4. Ivanov V.I., Rudomazina Yu.D., 2005, “About Tur´an problem for periodic functions with nonnegative Fourier coefficients and small support”, Math. Notes, vol. 77, № 6, pp. 870–875.

5. Ivanov V.I., Gorbachev D.V., Rudomazina Yu.D., 2005, “Some extremal problems for periodic functions with conditions on their values and Fourier coefficients”, Proc. Steklov Inst. Math., № 2 suppl, pp. 139–159.

6. Ivanov V.I., 2006, “On the Tur´an and Delsarte problems for periodic positive definite functions”, Math. Notes, vol. 80, № 6, pp. 875–880.

7. Ivanov V.I., Ivanov A.V., 2010, “Tur´an problems for periodic positive definite functions”, Annales Univ. Sci. Budapest, Sect. Comp., vol. 33, pp. 219–237.

8. Andreev N.N., 1997, “An extremal problem for periodic functions with small support”, Moscow Univ. Math. Bull., vol 52, pp. 29–32.

9. Gorbachev D.V., 2001, “An extremal problem for periodic functions with supports in the ball”, Math. Notes, vol. 69, № 3, pp. 313–319.

10. Siegel C.L., 1935, “¨Uber Gitterpunkte in konvexen K¨orpern und damit zusammenh¨angendes Extremal problem”, Acta Math., vol. 65, № 1, pp. 307–323.

11. Boas R.P., Kac M., 1945, “Inequalities for Fourier Transforms of positive functions”, Duke Math. J., vol. 12, pp. 189–206.

12. Kolountzakis M.N., R´ev´esz Sz.Gy., 2003, “On a problem of Tura´n about positive definite functions”, Proc. Amer. Math. Soc., vol. 131, pp. 3423–3430.

13. Arestov V.V., Berdysheva E.E., 2001, “Tur´an’s problem for positive definite functions with supports in a hexagon”, Proc. Steklov Inst. Math., № 1, suppl, pp. 20–29.

14. Arestov V.V., Berdysheva E.E., 2002, “The Tur´an problem for a class of polytopes”, East J. Approx., vol. 8, № 3, pp. 381–388.

15. Kolountzakis M.N., R´ev´esz Sz.Gy., 2006, “Tur´an’s extremal problem for positive definite functions on groups”, J. London Math. Soc., vol. 74, pp. 475–496.

16. R´ev´esz Sz.Gy., 2011, “Tur´an’s extremal problem on locally compact abelian groups”, Anal. Math., vol. 37, № 1, pp. 15–50.

17. Viazovska M.S., 2017, “The sphere packing problem in dimension 8”, Annals of Math., vol. 185, № 3, pp. 991–1015.

18. Cohn H., Kumar A., Miller S.D., Radchenko D., Viazovska M., 2016, “The sphere packing problem in dimension 24”, Annals of Math., vol. 185, № 3, pp. 1017–1033.

19. Gorbachev D.V., 2000, “An extremal problem for an entire functions of exponential spherical type related to the Levenshtein estimate for the sphere packing density in Rn”, Izv. Tul. Gos. Univ., Ser. Mat. Mec. Inf., vol. 6, № 1, pp. 71–78. [in Russian]

20. Cohn H., 2002, “New upper bounds on sphere packings II”, Geom. Topol., vol. 6, pp. 329–353.

21. Levenshtein V.I., 1979, “Bounds for packings in n-dimensional Euclidean space”, Soviet Math. Dokl., vol. 20, pp. 417–421.

22. Yudin V.A., 1989, “Packings of balls in Euclidean space, and extremal problems for trigonometric polynomials”, Discrete Math. Appl., vol. 1, № 1, pp. 69–72.

23. Yudin V.A., 1976 “The multidimensional Jackson theorem”, Math. Notes, vol. 20, № 3, pp. 801– 804.

24. Ivanov V.I., 1994 “Approximation of functions in spaces Lp”, Math. Notes, vol. 56, № 2, pp. 770–789.

25. Bohman H., 1960, “Approximate Fourier analysis of distribution functions”, Ark. Mat., vol. 4, pp. 99–157.

26. Ehm W., Gneiting T., Richards D., 2004, “Convolution roots of radial positive definite functions with compact support”, Trans. Amer. Math. Soc., vol. 356, pp. 4655–4685.

27. Logan B.F., 1983, “Extremal problems for positive-definite bandlimited functions. I. Eventually positive functions with zero integral”, SIAM J. Math. Anal., vol. 14, № 2, pp. 249–252.

28. Logan B.F., 1983, “Extremal problems for positive-definite bandlimited functions II. Eventually negative functions”, SIAM J. Math. Anal., vol. 14, № 2, pp. 253–257.

29. Gorbachev D.V., 2000, “Extremum problems for entire functions of exponential spherical type”, Math. Notes, vol. 68, № 2, pp. 159–166.

30. Chernykh N.I., 1967, “On Jackson’s inequality in L2”, Proc. Steklov Inst. Math., vol. 88, pp. 75– 78.

31. Yudin V.A., 1981, “Multidimensional Jackson theorem in L2”, Math. Notes, vol. 29, № 2, pp. 158–162.

32. Berdysheva E.E., 1999, “Two related extremal problems for entire functions of several variables”, Math. Notes, vol. 66, № 3, pp. 271–282.

33. Arestov V.V., Chernykh N.I., 1981, “On the L2-approximation of periodic functions by trigonometric polynomials”, Approximation and functions spaces: Proc. intern. conf., Gdansk, 1979. Amsterdam: North-Holland, pp. 25—43.

34. Ivanov A.V., Ivanov V.I., 2013, “Optimal arguments in Jackson’s inequality in the powerweighted space L2(Rd)”, Math. Notes, vol. 94, № 3, pp. 320–329.

35. Ivanov A.V., 2010, “Some extremal problem for entire functions in weighted spaces”, Izv. Tul. Gos. Univ., Estestv. Nauki, № 1, pp. 26–44. [in Russian]

36. Ivanov A.V., 2011, “Logan problem for entire functions of several variables and Jackson constants in weighted spaces”, Izv. Tul. Gos. Univ., Estestv. Nauki, № 2, pp. 29–58. [in Russian]

37. Gorbachev D.V., Ivanov V.I., 2017, “Boman extremal problem for Dunkl transform”, Proc. Steklov Inst. Math., vol 297, № 1 suppl., pp. 88–96.

38. Platonov S.S., 2007, “Bessel harmonic analysis and approximation of functions on the half-line”, Izvestiya: Mathematics, vol 71, № 5, pp. 1001–1048.

39. Gorbachev D.V., 2014, “Boman extremal problem for Fourier–Hankel transform”, Izv. Tul. Gos. Univ., Estestv. Nauki, № 4, pp. 5–10. [in Russian]

40. Gorbachev D.V., 2005, “Selected Problems in the Theory of Functions and Approximation Theory: Their Applications”, Tula: Grif and K, 192 p. [in Russian]

41. Frappier C., Olivier P., 1993, “A quadrature formula involving zeros of Bessel functions”, Math. Comp., vol. 60, pp. 303–316.

42. Grozev G.R., Rahman Q.I., 1995, “A quadrature formula with zeros of Bessel functions as nodes”, Math. Comp., vol. 64, pp. 715–725.

43. Ghanem R.B., Frappier C., 1998, “Explicit quadrature formulae for entire functions of exponential type”, J. Approx. Theory, vol. 92, № 2, pp. 267–279.

44. Gorbachev D.V., Ivanov V.I., 2015, “Gauss and Markov quadrature formulae with nodes at zeros of eigenfunctions of a Sturm–Liouville problem, which are exact for entire functions of exponential type”, Sbornik: Math., vol. 206, № 8, pp. 1087–1122.

45. Flensted-Jensen M., Koornwinder T.H., 1973, “The convolution structure for Jacobi function expansions”, Ark. Mat., vol. 11, pp. 245–262.

46. Flensted-Jensen M., Koornwinder T.H., 1979, “Jacobi functions: The addition formula and the positivity of dual convolution structure”, Ark. Mat., vol. 17, pp. 139–151.

47. Gorbachev D.V., Ivanov V.I., 2017, “Tur´an’s and Fej´er’s extremal problems for Jacobi transform”, Anal. Math. [In press]

48. Gorbachev D.V., Ivanov V.I., Smirnov O.I., 2016, “The Delsarte Extremal Problem for the Jacobi Transform”, Math. Notes, vol. 100, № 5, pp. 677–686.

49. Gorbachev D.V., Ivanov V.I., 2016, “Boman extremal problem for Jacobi transform”, Trudy Inst. Mat. Mekh. UrO RAN, vol. 22, № 4, pp. 126–135. [in Russian]

50. Gorbachev D.V., Ivanov V.I., 2017, “Some extremal problems for Fourier transform on hyperboloid”, Math. Notes, vol. 102, № 4, pp. 480–491.

51. Gorbachev D.V., Ivanov V.I., Veprintsev R.A., 2014, “Optimal Argument in Sharp Jackson’s inequality in the Space L2 with the Hyperbolic Weight”, Math. Notes, vol. 96, № 6, pp. 338–348.

52. Veprintsev R.A., 2015, “Approximation in L2 by partial integrals of the multidimensional Jacobi transform”, Math. Notes, vol. 97, № 6, pp. 815–831.

53. Gorbachev D.V., Ivanov V.I., 2016, “Approximation in L2 by partial integrals of the Fourier transform over the eigenfunctions of the Sturm–Liouville operator”, Math. Notes, vol. 100, № 4, pp. 540–549.

54. Gorbachev D.V., Ivanov V.I., Veprintsev R.A., 2016, “Approximation in L2 by partial integrals of the multidimensional Fourier transform over the eigenfunctions of the Sturm–Liouville operator”, Trudy Inst. Mat. Mekh. UrO RAN, vol. 22, № 4, pp. 136–152. [in Russian]

55. Gorbachev D.V., Ivanov V.I., 2017, “Some extremal problems for the Fourier transform over the eigenfunctions of the Sturm–Liouville operator”, Chebyshevskii sbornik, vol. 18, № 2, pp. 34–53. [in Russian]

56. Vilenkin N.J., 1968, “Special Functions and the Theory of Group Representations”, Providence, RI: Amer. Math. Soc., Translations of mathematical monographs, vol. 22, 613 p.

57. Lizorkin P.I., 1992, “Direct and inverse theorems of approximation theory for functions on the Lobachevsky space”, Proc. Steklov Inst. Math., vol. 194. pp. 125–151.

58. Gorbachev D.V., Piskorsh M.S., 1998, “Sharp Jackson inequality in L2 on hyperboloid”, Izv. Tul. Gos. Univ., Ser. Mat. Mec. Inf., vol. 4, № 1. pp. 54–58. [in Russian]

59. Popov V.Yu., 1998, “Sharp Jackson–Stechkin inequality in L2 on the hyperboloid”, Trudy Inst. Mat. Mekh. UrO RAN, vol. 5, № 4. pp. 254–266. [in Russian]

60. Petrova I.V., 1992, “Approximation on the hyperboloid in L2-metric”, Proc. Steklov Inst. Math., vol. 194. pp. 229–243.