Abstract: As the partial sum operator S^(N,B)_n(f;Z) of Neumann-Bessel series can not uniformly converge for each continuous f(Z) on unit circle Γ, in order to improve the convergence the operator of inter polation polynomial, the kernel function K^(N,B)_n(Z,ξ) of Neumann-Besssl series was divided by 2 to construct a new Rogosinski kernel and it has been proved in detail that such a new operator uniformly converges for any continuous function f(Z) on the unit circle |Z|=1 and has the best approxi mation order for f(Z) on |Z|=1.

Key words: Neumann-Bessel series, kernel functions, uniformlyconvergent, the best approximation order