The second Logan extremal problem for the fourier transform over the eigenfunctions of the Sturm--Liouville operator

D. V. Gorbachev ; V. I. Ivanov ; E. P. Ofitserov ; O. I. Smirnov

Čebyševskij sbornik, Tome 19 (2018) no. 1, pp. 57-78.

Résumé

For the cosine Fourier transform on the half-line two extremal problems were posed and solved by B. Logan in 1983. In the first problem it was necessary to find a minimal neighborhood of zero outside of which a nontrivial integrable even entire function of exponential type at most $\tau$, having a nonnegative Fourier transform, is nonpositive. In the second problem it was necessary to find a minimal neighborhood of zero outside of which a nontrivial integrable even entire function of exponential type at most $\tau$, having a nonnegative Fourier transform and a zero mean value, is nonnegative. The first Logan problem got the greatest development, because it turned out to be connected with the problem of the optimal argument in the modulus of continuity in the sharp Jackson inequality in the space $L^2$ between the value of the best approximation of function by entire functions of exponential type and its modulus of continuity. It was solved for the Fourier transform on Euclidean space and for the Dunkl transform as its generalization, for the Fourier transform over eigenfunctions of the Sturm–Liouville problem on the half-line, and the Fourier transform on the hyperboloid. The second Logan problem was solved only for the Fourier transform on Euclidean space. In the present paper, it is solved for the Fourier transform over eigenfunctions of the Sturm-Liouville problem on the half-line, in particular, for the Hankel and Jacobi transforms. As a consequence of these results, using the averaging method of functions over the Euclidean sphere, we obtain a solution of the second Logan problem for the Dunkl transform and the Fourier transform on the hyperboloid. General estimates are obtained using the Gauss quadrature formula over the zeros of the eigenfunctions of the Sturm–Liouville problem on the half-line, which was recently proved by the authors of the paper. In all cases, extremal functions are constructed. Their uniqueness is proved.

Détail

Keywords: The Sturm–Liouville problem on semidirect, Fourier transform on semidirect, Dunkl transformation, Fourier transform on a hyperboloid, extremal Logan's problems, Gaussian quadrature formula.

Bibliographie

[1] 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

[2] 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

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

[4] Chernykh N. I., “On Jackson's inequality in $L_2$”, Proc. Steklov Inst. Math., 88 (1967), 75–78 | MR | Zbl | Zbl

[5] Yudin V. A., “The multidimensional Jackson theorem”, Math. Notes, 20:3 (1976), 801–804 | DOI | MR | Zbl

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

[7] 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)

[8] Gorbachev D. V., Ivanov V. I. “Some extremal problems for Fourier transform on hyperboloid”, Math. Notes, 102:4 (2017), 480–491 | MR | Zbl

[9] Gorbachev D. V., Ivanov V. I., Ofitserov E. P., Smirnov O. I., “Some extremal problems of harmonic analysis and approximation theory”, Chebyshevskii Sbornik, 18:4 (2017), 139–166 (in Russian) | DOI

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

[11] 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

[12] 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

[13] Gorbachev D. V., Ivanov V. I., “Some extremal problems for the Fourier transform over eigenfunctions of the Sturm–Liouville operator”, Chebyshevskii Sbornik, 18:4 (2017), 139–166 (in Russian) | DOI | Zbl

[14] Arestov V. V., Chernykh N. I., “On the $L_2$-approximation of periodic functions by trigonometric polynomials”, Approximation and functions spaces, Proc. intern. conf. (Gdansk, 1979), North-Holland, Amsterdam, 1981, 25–43 | MR

[15] Ivanov A. V., Ivanov V. I., “Optimal arguments in Jackson's inequality in the power-weighted space $L_2(\mathbb{R}^d)$”, Math. Notes, 94:3 (2013), 320–329 | DOI | DOI | MR | Zbl

[16] 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

[17] Veprintsev R. A., “Approximation in $L_2$ by partial integrals of the multidimensional Jacobi transform”, Math. Notes, 97:6 (2015), 815–831 | DOI | Zbl

[18] 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

[19] 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)

[20] Gorbachev D. V., Strankovskii S. A., “An extremal problem for even positive definite entire functions of exponential type”, Math. Notes, 80:5 (2006), 673–678 | DOI | DOI | MR | Zbl

[21] Ivanov V. I., Ivanov A. V., “Optimal arguments in the Jackson-Stechkin inequality in $L_2(\mathbb{R}^d)$ with Dunkl weight”, Math. Notes, 96:5 (2014), 666–677 | DOI | DOI | MR | Zbl

[22] Ivanov V., Ivanov A., “Generalized Logan's Problem for Entire Functions of Exponential Type and Optimal Argument in Jackson's Inequality in $L_2(\mathbb{R}^3)$”, Acta. Math. Sin., English Ser. (First Online: 28 April 2018) | MR

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

[24] Yudin V. A., “Disposition of Points on a Torus and Extremal Properties of Polynomials”, Proc. Steklov Inst. Math., 219 (1997), 447–457 | MR | Zbl

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

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

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

[28] 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

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

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

[31] Rösler M., “A positive radial product formula for the Dunkl kernel”, Trans. Amer. Math. Soc., 355:6 (2003), 2413–2438 | DOI | MR | Zbl

[32] Rösler M., “Dunkl Operators: Theory and Applications”, Lecture Notes in Math., 1817, Springer, Berlin, 2003, 93–135 | DOI | MR | Zbl

[33] de Jeu M., “Paley–Wiener theorems for the Dunkl transform”, Trans. Amer. Math. Soc., 358:10 (2006), 4225–4250 | DOI | MR | Zbl

[34] Xu Y., “Dunkl operators: Funk-Hecke formula for orthogonal polynomials on spheres and on balls”, Bull. London Math. Soc., 32 (2000), 447–457 | DOI | MR | Zbl

[35] Vilenkin N. J., Special Functions and the Theory of Group Representations, Translations of mathematical monographs, 22, Amer. Math. Soc., Providence, RI, 1968, 613 pp. | DOI | MR | Zbl

[36] Koornwinder T. H., “A new proof of a Paley–Wiener type theorem for the Jacobi transform”, Ark. Mat., 13 (1979), 145–159 | DOI | MR