Some extremal problems for the Fourier transform over the eigenfunctions of the Sturm--Liouville operator

D. V. Gorbachev ; V. I. Ivanov

Čebyševskij sbornik, Tome 18 (2017) no. 2, pp. 34-53.

Résumé

The Turán, Fejér, Delsarte, Bohman, and Logan extremal problems for positive definite functions in Euclidean space or for functions with nonnegative Fourier transform have many applications in the theory of functions, approximation theory, probability theory, and metric geometry. Since the extremal functions in them are radial, by means of averaging over the Euclidean sphere they admit a reduction to analogous problems for the Hankel transform on the half-line. For the solution of these problems we can use the Gauss and Markov quadrature formulae on the half-line at zeros of the Bessel function, constructed by Frappier and Olivier. The normalized Bessel function, as the kernel of the Hankel transform, is the solution of the Sturm–Liouville problem with power weight. Another important example is the Jacobi transform, the kernel of which is the solution of the Sturm–Liouville problem with hyperbolic weight. The authors of the paper recently constructed the Gauss and Markov quadrature formulae on the half-line at zeros of the eigenfunctions of the Sturm–Liouville problem under natural conditions on the weight function, which, in particular, are satisfied for power and hyperbolic weights. Under these conditions on the weight function, the Turán, Fejér, Delsarte, Bohman, and Logan extremal problems for the Fourier transform over eigenfunctions of the Sturm–Liouville problem are solved. Extremal functions are constructed. For the Turán, Fejér, Bohman, and Logan problems their uniqueness is proved. Bibliography: 44 titles.

Détail

Keywords: Sturm–Liouville problem on the half-line, Fourier transform, Turán, Fejér, Delsarte, Bohman and Logan extremal problems, Gauss and Markov quadrature formulae.

Bibliographie

[1] Arestov V. V., Berdysheva E. E., “Turán's problem for positive definite functions with supports in a hexagon”, Proc. Steklov Inst. Math., 2001, no. 1, suppl., 20–29 | MR

[2] Arestov V. V., Berdysheva E. E., “The Turán problem for a class of polytopes”, East J. Approx., 8:3 (2002), 381–388 | MR | Zbl

[3] Berdysheva E. E., “Two related extremal problems for entire functions of several variables”, Math. Notes, 66:3 (1999), 271–282 | DOI | DOI | MR | Zbl

[4] Boas R. P., Kac M., “Inequalities for Fourier Transforms of positive functions”, Duke Math. J., 12 (1945), 189–206 | DOI | MR | Zbl

[5] Bohman H., “Approximate Fourier analysis of distribution functions”, Ark. Mat., 4 (1960), 99–157 | DOI | MR

[6] Cohn H., “New upper bounds on sphere packings II”, Geom. Topol., 6 (2002), 329–353 | DOI | MR | Zbl

[7] Ehm W., Gneiting T., Richards D., “Convolution roots of radial positive definite functions with compact support”, Trans. Amer. Math. Soc., 356 (2004), 4655–4685 | DOI | MR | Zbl

[8] Gorbachev D. V., “Extremum problems for entire functions of exponential spherical type”, Math. Notes, 68:2 (2000), 159–166 | DOI | DOI | MR | Zbl

[9] Gorbachev D. V., “An extremal problem for an entire functions of exponential spherical type related to the Levenshtein estimate for the sphere packing density in $\mathbb{R}^n$”, Izv. Tul. Gos. Univ., Ser. Mat. Mekh. Inform., 6:1 (2000), 71–78 (in Russian)

[10] Gorbachev D. V., “An extremal problem for periodic functions with supports in the ball”, Math. Notes, 69:3 (2001), 313–319 | DOI | DOI | MR | Zbl

[11] Kolountzakis M. M., Révész Sz. Gy., “On a problem of Turán about positive definite functions”, Proc. Amer. Math. Soc., 131 (2003), 3423–3430 | DOI | MR | Zbl

[12] Kolountzakis M. M., Révész Sz. Gy., “Turán's extremal problem for positive definite functions on groups”, J. London Math. Soc., 74 (2006), 475–496 | DOI | MR | Zbl

[13] Logan B. F., “Extremal problems for positive-definite bandlimited functions. I. Eventually positive functions with zero integral”, SIAM J. Math. Anal., 14:2 (1983), 249–252 | DOI | MR | Zbl

[14] Logan B. F., “Extremal problems for positive-definite bandlimited functions. II. Eventually negative functions”, SIAM J. Math. Anal., 14:2 (1983), 253–257 | DOI | MR | Zbl

[15] Révész Sz. Gy., “Turán's extremal problem on locally compact abelian groups”, Anal. Math., 37:1 (2011), 15–50 | DOI | MR

[16] Siegel C. L., “Über Gitterpunkte in konvexen Körpern und damit zusammenhängendes Extremal problem”, Acta Math., 65 (1935), 307–323 | DOI | MR

[17] Fejér L., “Über trigonometrische Polynome”, J. Angew. Math., 146 (1915), 53–82 | MR | Zbl

[18] Gorbachev D. V., Manoshina A. S., “Turán Extremal Problem for Periodic Functions with Small Support and Its Applications”, Math. Notes, 76:5 (2004), 640–652 | DOI | DOI | MR | Zbl

[19] Ivanov V. I., Rudomazina Yu. D., “About Turán problem for periodic functions with nonnegative Fourier coefficients and small support”, Math. Notes, 77:6 (2005), 870–875 | DOI | DOI | MR | Zbl

[20] Ivanov V. I., Gorbachev D. V., Rudomazina Yu. D., “Some extremal problems for periodic functions with conditions on their values and Fourier coefficients”, Proc. Steklov Inst. Math., 2005, no. 2, suppl., 139–159 | MR | Zbl

[21] Ivanov V. I., “On the Turán and Delsarte problems for periodic positive definite functions”, Math. Notes, 80:6 (2006), 875–880 | DOI | DOI | MR | Zbl

[22] Ivanov V. I., Ivanov A. V., “Turán problems for periodic positive definite functions”, Annales Univ. Sci. Budapest, Sect. Comp., 33 (2010), 219–237 | MR | Zbl

[23] Stechkin S. B., “An extremal problem for trigonometric series with nonnegative coefficients”, Acta Math. Acad. Sci. Hung., 23:3–4 (1972), 289–291 | MR | Zbl

[24] Gorbachev D. V., Selected Problems in the Theory of Functions and Approximation Theory: Their Applications, Grif and K, Tula, 2005, 192 pp. (in Russian)

[25] Gorbachev D. V., “Boman extremal problem for Fourier–Hankel transform”, Izv. Tul. Gos. Univ., Ser. Estestv. Nauki, 2014, no. 4, 5–10 (in Russian)

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

[27] Gorbachev D. V., Ivanov V. I., “Boman extremal problem for Dunkl transform”, Trudy Inst. Mat. Mekh. UrO RAN, 21, no. 4, 2015, 115–123 (in Russian)

[28] Frappier C., Olivier P., “A quadrature formula involving zeros of Bessel functions”, Math. Comp., 60 (1993), 303–316 | DOI | MR | Zbl

[29] Grozev G. R., Rahman Q. I., “A quadrature formula with zeros of Bessel functions as nodes”, Math. Comp., 64 (1995), 715–725 | DOI | MR | Zbl

[30] Gorbachev D. V., Ivanov V. I., “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., 206:8 (2015), 1087–1122 | DOI | DOI | MR | Zbl

[31] Gorbachev D. V., Ivanov V. I., Smirnov O. I., “The Delsarte Extremal Problem for the Jacobi Transform”, Math. Notes, 100:5 (2016), 677–686 | DOI | MR | Zbl

[32] Gorbachev D. V., Ivanov V. I., “Boman extremal problem for Jacobi transform”, Trudy Inst. Mat. Mekh. UrO RAN, 22, no. 4 (2016), 126–135 (in Russian)

[33] Gorbachev D. V., Ivanov V. I., “Turán's and Fejér's extremal problems for Jacobi transform”, Anal. Math., 2017 (to appear)

[34] Gorbachev D. V., Ivanov V. I., Smirnov O. I., “Some extremal problems for Fourier transform on hyperboloid”, Math. Notes, 2017 (to appear)

[35] Levitan B. M., Sargsyan I. S., Introduction to spectral theory, Nauka, M., 1970, 671 pp. (in Russian)

[36] Levitan B. M., Sargsyan I. S., Sturm–Liouville and Dirac Operators, Nauka, M., 1988, 432 pp. (in Russian)

[37] Gorbachev D. V., Ivanov V. I., “Approximation in $L_2$ by partial integrals of the Fourier transform over the eigenfunctions of the Sturm–Liouville operator”, Math. Notes, 100:4 (2016), 540–549 | DOI | DOI | MR | Zbl

[38] Gorbachev D. V., Ivanov V. I., Veprintsev R. A., “Approximation in $L_2$ by partial integrals of the multidimensional Fourier transform over the eigenfunctions of the Sturm–Liouville operator”, Trudy Inst. Mat. Mekh. UrO RAN, 22, no. 4 (2016), 136–152 (in Russian)

[39] Olver F. W. J., Introduction to asymptotics and special functions, Academic Press, New York, 1974, 297 pp. | MR | Zbl

[40] Flensted-Jensen M., Koornwinder T. H., “The convolution structure for Jacobi function expansions”, Ark. Mat., 11 (1973), 245–262 | DOI | MR | Zbl

[41] Flensted-Jensen M., Koornwinder T. H., “Jacobi functions: The addition formula and the positivity of dual convolution structure”, Ark. Mat., 17 (1979), 139–151 | DOI | MR | Zbl

[42] Levitan B. M., Theory of generalized translation operators, Nauka, M., 1973, 312 pp. (in Russian)

[43] Gorbachev D. V., Ivanov V. I., Veprintsev R. A., “Optimal Argument in Sharp Jackson's inequality in the Space $L_2$ with the Hyperbolic Weight”, Math. Notes, 96:6 (2014), 338–348 | MR

[44] Levin B. Ya., Distribution of Roots of Entire Functions, Gostekhizdat, M., 1956, 632 pp. (in Russian)