Polynomial approximation with an exponential weight on the real semiaxis

We consider the polynomial approximation on (0,+∞), with the weight u(x) = xγe-x-α-xβ, α > 0, β > 1 and γ ≧ 0. We introduce new moduli of smoothness and related K-functionals for functions defined on the real semiaxis, which can grow exponentially both at 0 and at +∞. Then we prove the Jackson theorem, also in its weaker form, and the Stechkin inequality. Moreover, we study the behavior of the derivatives of polynomials of best approximation. © 2013 Akadémiai Kiadó, Budapest, Hungary.