Abstract: In this paper，a new third type S．N．Bernstein interpolation polynomial H _n （ f ；x ，r ） with zeros of the Chebyshev ploynomial of the second kind is constructed．H n（ f ；x ，r） converge uniformly on ［－1，1］ for any continuous function f （ x ） ．T he convergence order is the best order if f （ x ） ∈C ^j ［－1，1］，0≤j ≤r，where r is an odd natural number．

Key words: uniform approximation；the best convergence order；Lagrange interpolation polynomial ,