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 \begin{equation*} \sum _{k=1}^{n}\Big (\frac {1}{k}\sum _{j=1}^{k}a_j\Big )^2\leq d_n \sum _{k=1}^{n}a_k^2, \qquad (a_1,\ldots ,a_n) \in \mathbb {R}^n \end{equation*} and \begin{equation*} \int _{0}^{\infty }\Bigg (\frac {1}{x}\int _{0}^{x}f(t) dt\Bigg )^2 dx \leq c_n \int _{0}^{\infty }f^2(x) dx,\qquad f\in \mathcal {H}_n, \end{equation*} for the finite dimensional spaces $\mathbb {R} ^n$ and $\mathcal {H}_n\colonequals \{f : \int _0^x f(t) dt =e^{-x/2} p(x)\ :\ p\in \mathcal {P}_n, p(0)=0\}$, 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 to be expressed in terms of the smallest zeros of the so-called continuous dual Hahn polynomials and the two-sided estimates for $d_n$ and $c_n$ of the form \begin{equation*} 4-\frac {c}{\ln n}< d_n, c_n<4-\frac {c}{\ln ^2 n} ,\qquad c>0 \end{equation*} are established.