Discrete uniformly convergent processes on
the roots of four kinds of Chebyshev polynomials

Abstract. Starting from discrete Fourier series we construct approximation processes on the roots of four kinds of Chebyshev polynomials generated by suitable summation functions \(\varphi\). We prove a general result stating that if the Fourier transform of \(\varphi\) is integrable then these processes are uniformly convergent on the whole interval \([-1,1]\) in some weighted spaces of continuous functions. We also examine necessary and sufficient conditions for the interpolation. As applications, we obtain various new results for the arithmetic means of the Lagrange interpolation, the Grünwald, the de la Vallée Poussin and the Hermite–Fejér interpolation.