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Miloslav Znojil
Reviewer number:
Item's zbl-Number:
DE 016 852 647
Lomov, I. S.:
The local convergence of biorthogonal series related to differential operators with nonsmooth coefficients. I.
Differ. Equ. 37, 351 - 366 (2001); translation from Differ. Uravn. 37, No. 3, 328 - 342 (2001).
47E05Ordinary differential operators
Primary Classification:
40A05Convergence and divergence of series and sequences
Secondary Classification:
34L10Eigenfunction expansions, completeness of eigenfunctions
42C99None of the above, but in this section
47A70Generalized eigenfunction expansions; rigged Hilbert spaces
46B15Summability and bases
non-selfadjoint ordinary linear differential operator of even order; set of the eigenfunctions and associated functions; biorthogonal set; partial sums of the biorthogonal series; partial sums of the Fourier trigonomertric series; equiconvergence estimates

Known results on the convergence rate of Fourier trigonometric
series may be transferred to the case of any bi-orthogonal expansion
via the so called equiconvergence theorem. In such a setting the
author contemplates a non-self-adjoint ordinary differential
operator L of order 2n and its so called root functions (i.e.,
eigenfunctions and the associated functions). Knowing that the
systems bi-orthogonal to these functions need not necessarily be
related to the adjoint operator (the existence of which need not
be guaranteed), the author continues his older analysis of the n=1
special case (reference [2]) and proves several auxiliary lemmas
and, partially, his main theorem on the equiconvergence rate of
the bi-orthogonal expansions in question. Several interesting
aspects of this type of problem are pointed out in the
introduction, and a few key ingredients of the proof (and, in
particular, an overall theorem on the local basis property and
some necessary estimates of integrals) are deferred to the part
two of the series.
Remarks to the editors:

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