A Criterion for the Uniform Convergence of the Lagrange–Jacobi Interpolation Process

We obtain a uniform and sufficient condition for the convergence of the Lagrange interpolation process with Jacobi nodes on a closed interval $[a,b]\subset(-1,1)$. The condition is stated in terms of the second differences of the interpolated function and uses its values only at the interpolation nodes. Some well-known criteria for uniform convergence are obtained as a consequence of our result.