Abstract: We constructed a kernel M_n(z,ξ) of Bernsteintype based on kernel function K_n(z,ξ) of NeumannBessel integral operator. The integral operator with the new kernel converges to any continuous functionf(z) on the unit circle Γ((|z|=1)uniformly, and has the best approximation order.

Key words: NeumannBessel series, kernel functions, uniformly convergent, best approximation order