  The paper deals with a special filtered approximation method, which
originates interpolation polynomials at Chebyshev zeros by using de la Vall\'ee
Poussin filters. These polynomials can be an useful device for many theoretical
and applicative problems since they combine the advantages of the classical
Lagrange interpolation, with the uniform convergence in spaces of locally
continuous functions equipped with suitable, Jacobi--weighted, uniform norms.
The uniform boundedness of the related Lebesgue constants, which equals to the
uniform convergence and is missing from Lagrange interpolation, has been
already proved in literature under different, but only sufficient, assumptions.
Here, we state the necessary and sufficient conditions to get it. These
conditions are easy to check since they are simple inequalities on the
exponents of the Jacobi weight defining the norm. Moreover, they are necessary
and sufficient to get filtered interpolating polynomials with a near best
approximation error, which tends to zero as the number $n$ of nodes tends to
infinity. In addition, the convergence rate is comparable with the error of
best polynomial approximation of degree $n$, hence the approximation order
improves with the smoothness of the sought function. Several numerical
experiments are given in order to test the theoretical results, to make a
comparison with the Lagrange interpolation at the same nodes and to show how
the Gibbs phenomenon can be strongly reduced.
