In commemoration of the 80th birthday anniversary of Professor Yongsheng SunWe research the simultaneous approximation problem of the quasi-Hermite interpolation that based on the zeros of Chebyshev polynomials under weighted Lp-norm where our estimation is sharp.Are the conditions in Theorem A necessary? It is so in some cases. For example, if w(x) = 1 √ 1−x 2 , m = 2, r = s = 0, then it is easy to verify that for Int. J. Wavelets Multiresolut Inf. Process. 2009.07:825-837. Downloaded from www.worldscientific.com by NANYANG TECHNOLOGICAL UNIVERSITY on 08/21/15. For personal use only.