\(L_p\)-inverse theorem for modified beta operators (Q1811905)

Beta operators are linear positive operators defined on \(L_p\) function spaces of the nonnegative real line. In this paper, an inverse result is established that derives a certain \(L_p\) modulus of smoothness from the order of approximation that is obtained when using beta operators. For this the approximand must be from \(L_p[0,\infty)\), the modulus of smoothness is of order \(2k+2\), and the approximation order is \(\alpha/2\) with \(0<\alpha<2k+2\).