Boundary value problems and related moduli of continuity (Q1916681)

The paper concerns expansions of functions from \(L_p[0, 1]\) in eigenfunctions and associated eigenfunctions of operators generated by ordinary differential expressions of order \(n\geq2\) with regular boundary conditions. In the general case of boundary conditions the rate of convergence of the expansion is estimated by modulus of continuity in \(L_p(\mathbb{R})\) of the function \(f\) extended by zero on \(\mathbb{R}\). The example shows that the estimation is in some sense sharp on the whole class of considered boundary conditions. For special types of boundary conditions the rate of convergence can be estimated by other moduli of continuity of the function \(f\). Proofs of the theorems are not presented.