On the equiconvergence rate with the Fourier integral of the spectral expansion associated with the self-adjoint extension of the Sturm-Liouville operator with uniformly locally integrable potential

In the space L2(ℝ), we consider the self-adjoint extension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{L} $$\end{document} of the Sturm-Liouville operator ly = −y″ + q(x)y whose potential q is uniformly locally integrable on ℝ, i.e., satisfies the condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \omega _q (h) = \mathop {\sup }\limits_{x \in \mathbb{R}} \int\limits_x^{x + h} {\left| {q(t)} \right|dt < + \infty ,h > 0.} $$\end{document}. We study the problem on the equiconvergence rate of the spectral expansion associated with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{L} $$\end{document} of a function f ∈ L1(ℝ) with the Fourier integral on the entire real line. We obtain uniform estimates of the equiconvergence rate under some additional conditions on f or q.