A generalization of the Whittaker-Kotel'nikov-Shannon sampling theorem for continuous functions on a closed interval

Classes of functions in the space of continuous functions $f$ defined on the interval $[0,\pi]$ and vanishing at its end-points are described for which there is pointwise and approximate uniform convergence of Lagrange-type operators
$$ S_\lambda(f,x)=\sum_{k=0}^n\frac{y(x,\lambda)}{y'(x_{k,\lambda}) (x-x_{k,\lambda})}f(x_{k,\lambda}). $$
These operators involve the solutions $y(x,\lambda)$ of the Cauchy problem for the equation
$$ y''+(\lambda-q_\lambda(x))y=0 $$
where $q_\lambda\in V_{\rho_\lambda}[0,\pi]$ (here $V_{\rho_\lambda}[0,\pi]$ is the ball of radius $\rho_\lambda=o(\sqrt\lambda/\ln\lambda)$ in the space of functions of bounded variation vanishing at the origin, and $y(x_{k,\lambda})=0$). Several modifications of this operator are proposed, which allow an arbitrary continuous function on $[0,\pi]$ to be approximated uniformly.
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