Hardy's inequalities in finite dimensional Hilbert spaces

Abstract

We study the behaviour of the smallest possible constants $d_n$ and $c_n$ in Hardy's inequalities $$\sum _{k=1}^n(\frac{1}{k}\sum _{j=1}^k{a}_j{)}^2\le d_n\sum _{k=1}^n{a}_k^2,(a_1,\dots ,a_n)\in {\mathbb{R}}^n$$ and $${\int }_0^{\infty }(\frac{1}{x}\underset{0}{\overset{x}{\int }}f(t)dt{)}^{2}dx\le c_n{\int }_0^{\infty }{f}^2(x)dx,f\in {\mathcal{H}}_n,$$ for the finite dimensional spaces ${\mathbb{R}}^n$ and ${\mathcal{H}}_n:=\{f:{\int }_0^{x}f(t)dt=e^{-x/2}p(x):p\in {\mathcal{P}}_n,p(0)=0\}$, where ${\mathcal{P}}_n$ is the set of real-valued algebraic polynomials of degree not exceeding $n$. The constants $d_n$ and $c_n$ are identified as the smallest eigenvalues of certain Jacobi matrices and the two-sided estimates for $d_n$ and $c_n$ of the form $$4-\frac{c}{\ln n}<d_n,c_n<4-\frac{c}{{\ln }^{2}n},c>0$$ are established.