Uniform convergence of Lagrange interpolation based on the Jacobi nodes (Q1125968)

Lagrange interpolation and its related variations are of great mathematical interest in several contexts. Particularly the problem of its uniform convergence based on specified nodes is very important. Here, the author has given the necessary and sufficient conditions for a continuous function guaranteeing the uniform convergence on the interval \([-1,1]\) of its Lagrange interpolant based on the Jacobi nodes. The conditions are in terms of \(\Lambda\)-variation, \(\Phi\)-variation, modulus variation and Banach indicatrix of functions. After introducing the special notations six definitions, five theorems and five lemmas are presented. Proofs to the theorems are also provided in detail. A comprehensive list of relevant references attached makes the study very helpful.