Best Approximation of Functions like |x|λ exp(−A|x|−α) Original Research Article

We determine the exact order of best approximation by polynomials and entire functions of exponential type of functions likeϕλ, α(x)=|x|λ exp(−A|x|−α). In particular, it is shown thatE(ϕλ, α, Pn, Lp(−1, 1))∼n−(2λp+αp+2)/2p(1+α)×exp(−(1+α−1)(Aα)1/(1+α) cos απ/2(1+α) nα/(1+α)), whereE(ϕλ, α, Pn, Lp(−1, 1)) denotes best polynomial approximation ofϕλ, αinLp(−1, 1),λ∈R,α∈(0, 2],A>0, 1⩽p⩽∞. The problem, concerning the exact order of decrease ofE(ϕ0, 2, Pn, L∞(−1, 1)), has been posed by S. N. Bernstein.