On the Measure of Approximation for Some Linear Means of Trigonometric Fourier Series Original Research Article

Many approximation methods inC2πmay be generated via a certain functionϕ∈C[0, 1]withϕ(0)=1,ϕ(1)=0. The functionϕj(t)=cos(j−1/2) πt(j∈N) generates the Rogosinski approximation method [N. K. Bari, “A Treatise on Trigonometric Series,” Vols. I, II, Pergamon Press, New York, 1964]. Our idea consists in representingϕby the orthogonal systemϕjto extend results previously known for the Rogosinski method to arbitrary approximation methods. We illustrate this by proving two asymptotic estimates for the measure of approximation.