Starting from the Lagrange interpolation on the roots of Jacobi polynomials, a wide class of discrete linear processes is
constructed using summations. Some special cases are also considered, such as the Fejr, de la Valle Poussin, Cesro, Riesz
and Rogosinski summations. The aim of this note is to show that the sequences of this type of polynomials are uniformly convergent
on the whole interval [-1,1] in suitable weighted spaces of continuous functions. Order of convergence will also be investigated.
Some statements of this paper can be obtained as corollaries of our general results proved in .