Equiconvergence of eigenfunction expansions for Sturm-Liouville operators with a distributional potential. (English. Russian original) Zbl 1209.34108
Sb. Math. 201, No. 9, 1307-1322 (2010); translation from Mat. Sb. 201, No. 9, 61-76 (2010).
Summary: The Sturm-Liouville operator \( L=-d^2/dx^2+q(x)\) in the space \( L_2 [ 0,\pi ] \) under Dirichlet boundary conditions is investigated. It is assumed that \(q(x)=u'(x)\), \(u(x)\in L_2 [ 0,\pi ] \) (here, differentiation is used in the distributional sense). The problem of when the expansion of a function \(f\) in terms of a series of eigenfunctions and associated functions of the operator \( L\) is uniformly equiconvergent on the whole of the interval \( [ 0,\pi ] \) with its Fourier sine series expansion is considered. It is shown that such uniform convergence holds for any function \(f\) in the space \( L_2 [ 0,\pi ] \).
MSC:
| 34L10 | Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
