The best approximation, which was initiated by Chebyshev in 1853, is an elegant idea in the area of polynomial approximations and every textbook on approximation theory has a chapter to introduce it. From the point of view of polynomial approximations in the maximum norm, there is no doubt that the best approximation is better than any other polynomial approximations of the same degree. However, its implementation is a nonlinear problem and the cost is prohibitively expensive when the degree is large. In this talk, we compare the convergence behavior of classical orthogonal projections, such as Legendre, Gegenbauer and Jacobi projections, and the best approximation. We show that the convergence rates of both Legendre and Chebyshev projections are almost the same as that of best approximations for analytic and differentiable functions. For Gegenbauer and Jacobi projections, however, their rates of convergence is the same as that of best approximations only under very restrictive conditions on the parameters of Gegenbauer and Jacobi polynomials. 

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