Hardy's inequalities in finite dimensional Hilbert spaces

We study the behaviour of the smallest possible constants d_n and c_n in Hardy's inequalities \displaystyle \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 and \displaystyle \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, 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 \displaystyle 4-\frac {c}{\ln n}< d_n, c_n0\, are established.