All Questions

Denote the set of real positive semidefinite $d\times d$ Hankel matrices as $\mathcal{S}$. Can we always decompose one $S\in \mathcal{S}$ into sum of rank $1$ $S_i\in \mathcal{S}$, i.e., $S=\sum_i S_i$...

Let $f_n(x)=\prod_{j=0}^{\lfloor{\frac{n-1}{2}}\rfloor}\prod_{i=2j+1}^{2n-2j-1}\frac{2x+i}{i}$ and $g_n(x)=\prod_{j=1}^{\lfloor{\frac{n}{2}}\rfloor}\prod_{i=2j}^{2n-2j}\frac{2x+i}{i}.$ Then $f_n(k)=\...

Let ${\left( {{a_n}} \right)_{n \geqslant 0}}$ be a sequence of real numbers and ${H_n} = \det \left( {{a_{i + j}}} \right)_{i,j = 0}^{n - 1}$ the $n-$th Hankel determinant. The sequence ${\left( {{...

Consider a sequence of real numbers $s=(s_0,s_1,\ldots)$. When is there a Borel measure $\mu$ supported on $[-1,1]$ so that $$ s_k = \int_{[-1,1]} x^k\,\mathrm{d}\mu,\quad \forall k\in\mathbb N\;? $$ ...