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.