The Hardy inequality and Birkhoff interpolation. (English) Zbl 0818.26010

Let \(U(x)\) denote the class of functions whose derivatives of order \(k- 1\) are absolutely continuous on \([0, 1]\) and which satisfy the boundary conditions \(U^{(i)}(0)= 0\), \(i\in M_ 0\), \(U^{(i)}(1)= 0\), \(j\in M_ 1\), where \(M_ i\subset (0, 1,\dots, k- 1)\), \(i= 0,1\). Suppose the two row incidence matrix of these boundary conditions satisfies the Pólya condition. The authors show that under these conditions the \(k\)th-order Hardy inequality is “meaningful”, viz. \[ \left( \int^ 1_ 0 | u(x)|^ q w_ 0(x) dx\right)^{1/q}\leq C\left( \int^ 1_ 0 | U^{(k)}(x)|^ p w_ k(x) dx\right) ^{1/p} \] with certain conditions on the parameters \(p,q> 1\).
[For further details, see the second author, Bayreuther Math. Schr. 44, 105-146 (1993; Zbl 0785.26010)].


26D15 Inequalities for sums, series and integrals
41A05 Interpolation in approximation theory


Zbl 0785.26010