Estimates of speed of convergence and equiconvergence of spectral decomposition of ordinary differential operators

I. S. Lomov

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 15 (2015) no. 4, pp. 405-418

Résumé

The present review contains results of V. A. Il'in and his pupils concerning an assessment of speed of convergence and equiconvergence with a trigonometrical series of Fourier of spectral decomposition of functions on root functions of linear ordinary differential operators both self-conjugate, and not self-conjugate, set on a final piece of a numerical straight line. The first theorem of V. A. Ilyin of equiconvergence of spectral decomposition for the differential operator of any order is provided. Theorems of the speed of equiconvergence of spectral decomposition at first for any self-conjugate expansions of the one-dimensional operator Schrodinger are formulated. Thus the potential of the operator can have any features on interval border. This allows us to receive new results even for all classical orthogonal polynomials. Further results for not self-conjugate operators are formulated. The review for the so-called loaded differential operators comes to the end with the theorem of equiconvergence speed. Estimates of speed of equiconvergence of decomposition are received both on any internal compact of an interval, and on the whole interval. Dependence of an assessment of speed of equiconvergence of decomposition on any compact of the main interval from distance of this compact to interval border is established.