Abstract:
We consider a linear ordinary differential operator defined by an $n$-th order differential expression with a nonzero coefficient for $(n-1)$th derivative and Birkhoff regular two-point boundary conditions. The question of the uniform convergence of expansions of a function into a series of root functions of the operator $L$ and the usual trigonometric Fourier series, as well as the estimation of the difference of the corresponding partial sums, is investigated. Estimates of the difference of the partial sums of these expansions are obtained in terms of the general (integral) modules of continuity of the expandable function and the coefficient at the $(n-1)$th derivative. The proof essentially uses the estimate (previously obtained by the author) of the difference between the partial sums of the expansions of a function in a series with respect to the root functions of the operator $L$ and in the modified trigonometric Fourier series, as well as the author's analogue of the Steinhaus theorem in terms of general modules of continuity.
Key words:ordinary differential operator, root functions, eigen and associated functions, expansion in a series by root functions, equiconvergence of expansions, estimation of the difference of partial sums, integral modules of continuity.
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