Convergence conditions for interpolation fractions at the nodes distinct from the singular points of a function

We consider an interpolation process for the class of functions with finitely many singular points by means of the rational functions whose poles coincide with the singular points of the function under interpolation. The interpolation nodes constitute a triangular matrix and are distinct from the singular points of the function. We find a necessary and sufficient condition for uniform convergence of sequences of interpolation fractions to the function under interpolation on every compact set disjoint from the singular points of the function and other conditions for convergence.