Abstract:
We consider a non-self-adjoint ordinary differential operator defined on a finite interval by an $n$th-order linear differential expression with a nonzero coefficient of the $(n-1)$th derivative and two-point Birkhoff regular boundary conditions. We study the uniform equiconvergence of expansions of a given function in a biorthogonal series in eigenfunctions and associated functions (or, briefly, root functions) of this operator and in an ordinary trigonometric Fourier series, as well as an estimate of the difference of the corresponding partial sums (or, briefly, the rate of equiconvergence) under the most general conditions on the expanded function and the coefficient of the $(n-1)$th derivative. We obtain estimates for the difference of the expansions in terms of general (integral) moduli of continuity of the expanded function and the coefficient of the $(n-1)$th derivative uniform inside the fundamental interval. From these estimates, corresponding estimates are derived in the case where moduli of continuity are bounded from above by slowly varying functions and, in particular, by logarithmic functions. Based on this, sufficient conditions for equiconvergence in the indicated cases are formulated. These results are obtained using the author's previously obtained estimate for the difference between the partial sums of expansions of a given function in a biorthogonal series in eigenfunctions and associated functions of the differential operator under consideration and in a modified trigonometric Fourier series, as well as analogues of the Steinhaus theorem. The modification of the trigonometric Fourier series consisted in applying a very specific bounded operator to the ordinary trigonometric Fourier series expressed through the coefficient of the $(n-1)$th derivative and its inverse operator to the expanded function.
Keywords:non-self-adjoint ordinary differential operator, expansion in root functions, Fourier series, equiconvergence of expansion, rate of equiconvergence.
Fulltext:
PDF file (404 kB)