## 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)].

### MSC:

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

### Keywords:

Birkhoff interpolation; Pólya condition; Hardy inequality

Zbl 0785.26010