Modification of even trigonometric interpolation polynomial

Aim An even trigonometric polynomial operator Wn(f:r,θ) is constructed(where r is a given natural number) based on these values of f(θ)(where f(θ)∈C2πand f(θ)) is even function) on these nodes {θk=knπ}nk=0.Methods The third method of Bernstein is used.Results The operator of Wn(f:r,θ) uniformly converge to f(θ) with 2π as its period on the total real axis,the convergence order of Wn(f:r,θ) reach the best convergence order when f(θ)∈Cj2π(0≤j≤r-1) and f(θ) is even functions.Conclusion Wn(f:r,θ) can uniformly converge to f(θ) on the total real axis on which the Lagrange operator is failed.