\(L_p\)-inverse theorem for modified beta operators. (English) Zbl 1080.41016

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 \(0<\alpha<2k+2\).


41A25 Rate of convergence, degree of approximation
41A30 Approximation by other special function classes
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