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.