The approximation of continuous periodic functions by means of extrapolation techniques for the De La Vall´ ee Poussin operator Francesco Costabile , Maria Italia Gualtieri , Stefano Serra November 1, 2013 Abstract We are concerned with the approximation of continuous 2π-periodic functions f by using trigonometric polynomials. It is well known that the De La Vall´ ee Poussin integral produces a trigonometric polyno- mial approximation V n (f ) which converges uniformly to the function f . Unfortunately, this technique is very slow and consequently is not employed for computational purposes. In this paper first we give a trigonometric version of the Taylor formula and then we prove an asymptotic expansion formula of V n (f ) which, joint with classical ex- trapolation procedures, allows one to approximate f fastly. Finally, we perform few numerical experiments which fully confirm the theoretical analysis. key words : De La Vall´ ee Poussin operator, extrapolation. AMS SC: 65D15, 65B05. 1 Introduction In this paper, we consider the use of De La Vall´ ee Poussin integrals [10] V n (f )= (2n)!! (2n − 1)!! 1 2π  π -π f (t) cos 2n (t − x) 2 dt (1) in order to approximate a given function belonging to the linear space of the continuous 2π functions C 2π . The theorem of Korovkin (see [9], pages 14-17) asserts that the conver- gence of positive linear operators generically defined in C θ = {f : ℜ→ℜ : f is continuous,f (x + θ)= f (x),x ∈ ℜ} 1