Exact constants in the inequalities between norms of functions and their derivatives
In connection with the study of the best approximation of the differentiation operator, \textit{V. N. Gabushin} [(*) ibid. 6, No.5, 573- 582 (1969; Zbl 0216.136)] has indicated a method of a construction of the best constant \(Q_{nk}\) for a Kolmogorov type inequality on the half line [0,\(\infty)\) with the mixed norms: \[ (1)\quad \| f^{(k)}\|_ c\leq Q_{nk}\| f\|_{L_ 2}^{1- (2k+1)/(2n)}\| f^{(n)}\|_{L_ 2}^{(2k+1)/(2n)}, \] where \(n=1,...,\infty\), \(k=0,...,n-1\), \(\| f\|_ C=\| f\|_{C[0,\infty)}\), \(\| f\|_{L_ 2}=\| f\|_{L_ 2[0,\infty)}\). A natural space of functions in which inequality (1) is considered is the space \(W_ 2^ n\) of functions f(t) absolutely continuous on [0,\(\infty)\) together with their derivatives up to the (n- 1)-th degree, and such that \(f^{(n)}\in L_ 2\). Changing slightly the notation of (*), we can describe the Gabushin method in the following way. At first it will be established that the following function is extremal for (1): \(g(t)=\sum^{n}_{v=1}a_ ve^{z_ vt}\), where \(z_{\nu}=ie^{i\pi (\nu -)/n}\), and the values \(a_ 1,a_ 2,...,a_ n\) are defined by the system of equations: \(\sum^{n}_{v=1}a_ vz_ v^ j=:\delta_{2n-k-1,j}\) \((j=1,...,2n-1)\). Next, the values of the quadratic forms \(A^ 2_ k=\| g\|_{L^ 2_ 2}\), \(B^ 2_ k=\| g^{(n)}\|_{L^ 2_ 2}\) are computed for the variables \(a_ 1,a_ 2,...,a_ n\), and \(Q_{nk}\) is calculated: \[ Q_{nk}=(A^ 2_ k+B^ 2_ k)/A_ k^{1-(2k+1)(2n)}B_ k^{(2k+1)/(2n)}. \] It is possible to calculate \(a_ v\) with the help of the Vieta formulas. In such a construction of \(Q_{nk}\) it is difficult to obtain any information on the behaviour of the value \(Q_{nk}\) depending on n and k. In the present note we develop another construction of \(Q_{nk}\), which allows us to show the property \(Q_{nk}=Q_{n,n-k-1}\) in an arbitrarily simple way, and to establish the asymptotics of \(Q_{nk}\) for a fixed k. For small values of k and an arbitrary n one can give explicit expressions for \(Q_{nk}\).
scientific article; zbMATH DE number 4069503