Equiconvergence of the Trigonometric Fourier Series and the Expansion in Eigenfunctions of the Sturm--Liouville Operator with a~Distribution Potential

I. V. Sadovnichaya

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 261 (2008), pp. 249-257

Résumé

We consider the Sturm–Liouville operator $L=-d^2/dx^2+q(x)$ with the Dirichlet boundary conditions in the space $L_2[0,\pi]$ under the assumption that the potential $q(x)$ belongs to $W_2^{-1}[0,\pi]$. We study the problem of uniform equiconvergence on the interval $[0,\pi]$ of the expansion of a function $f(x)$ in the system of eigenfunctions and associated functions of the operator $L$ and its Fourier sine series expansion. We obtain sufficient conditions on the potential under which this equiconvergence holds for any function $f(x)$ of class $L_1$. We also consider the case of potentials belonging to the scale of Sobolev spaces $W_2^{-\theta}[0,\pi]$ with $\frac12\theta\le1$. We show that if the antiderivative $u(x)$ of the potential belongs to some space $W_2^\theta[0,\pi]$ with $0\theta\frac12$, then, for any function in the space $L_2[0,\pi]$, the rate of equiconvergence can be estimated uniformly in a ball lying in the corresponding space and containing $u(x)$. We also give an explicit estimate for the rate of equiconvergence.