An inequality for continuous linear functionals (Q963815)

The main result of the paper is the inequality \(|A(f)| \leq M_k \|f^{(k)}\|_2\), where \(f\) has continuous \(n\)-th derivative over a closed interval, \(A\) is a continuous linear functional that annihilates all polynomials with degree less than \(n\), and \(f^{(k)}\) denotes the \(k\)-th derivative of \(f\), with \(k\) between \(2\) and \(n\) inclusively. The inequality implies that the absolute value of \(A(f)\) is controlled by the \(L_2\) norm of the \(k\)-th derivative of \(f\). For each \(k\), the sharp constant \(M_k\) is given, whose value is determined by the functional on a truncate power function. In the proof of the inequality, the functional \(A\) is applied to the Taylor's formula of \(f\), with the remainder expressed as an integral involving the product of the functional of a truncate power function and the \(k\)-th derivative of \(f\). Then Cauchy-Schwarz inequality is used to separate the functional of the truncate power function and the \(k\)-th derivative of \(f\) into two integrals. The constant \(M_k\) is obtained by estimating the integral of the functional of the truncate power function. Examples are given for the applications of the inequality with specifically selected functionals. These examples include estimating the integral of a function over a closed interval by the \(L_2\) norms of its derivatives, and estimating Lagrange interpolating polynomials for a function.