Abstract
One considers the spectral problem with boundary conditions , , for functions on . It is assumed that is a linear bounded operator from the Hölder space , , into and the are bounded linear functionals on with . Let be the linear span of the root functions of the problem , , , corresponding to the eigenvalues with , and let . An estimate of is obtained in terms of the -functional for (the direct theorem) and an estimate of this -functional is obtained in terms of for (the inverse theorem). In several cases two-sided bounds of the -functional are found in terms of appropriate moduli of continuity, and then the direct and the inverse theorems are stated in terms of moduli of continuity. For the spectral problem with periodic boundary conditions these results coincide with Jackson's and Bernstein's direct and inverse theorems on the approximation of functions by a trigonometric system.
Published Version
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