Geodesic
Equiconvergence of the Trigonometric Fourier Series and the Expansion in Eigenfunctions of the Sturm–Liouville Operator with a Distribution Potential
Informatics and Automation, Differential equations and dynamical systems, Tome 261 (2008), pp. 249-257 
I. V. Sadovnichaya. Equiconvergence of the Trigonometric Fourier Series and the Expansion in Eigenfunctions of the Sturm–Liouville Operator with a Distribution Potential. Informatics and Automation, Differential equations and dynamical systems, Tome 261 (2008), pp. 249-257. http://geodesic.mathdoc.fr/item/TRSPY_2008_261_a18/
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     author = {I. V. Sadovnichaya},
     title = {Equiconvergence of the {Trigonometric} {Fourier} {Series} and the {Expansion} in {Eigenfunctions} of the {Sturm{\textendash}Liouville} {Operator} with {a~Distribution} {Potential}},
     journal = {Informatics and Automation},
     pages = {249--257},
     year = {2008},
     volume = {261},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2008_261_a18/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

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