17 problems

Let a compact $m$-sheeted Riemann surface be partitioned into sheets by the Nuttall partition associated with a highlighted point. Denote the topmost closed sheet by the $m$th shee…

Let $f\in\mathbb C(z,w)$ and $f_\infty\in\mathscr H(\infty)$. Assume the setting of the subsection in which $E\subset\mathbb R$, $E\cap F=\varnothing$, and $F$ is symmetric with re…

Let $f\in\mathbb C(z,w)$, let $w_\infty$ be the specified element of the function $w$, and let $f_\infty\in\mathscr H(\infty)$ be the corresponding element of $f$. For each…

Let $Delta_1=[e_1,e_2]$ and $Delta_2=[e_3,e_4]$, where $e_1<e_2<e_3<e_4$. Consider the two-sheeted Riemann surface associated with the function $w^2=z^2-1$. Two-interval Nuttall-su…

Conjecture. Under these assumptions, assertion (A) of theorem M-1, namely the stated zero-distribution assertion for first-kind Hermite–Padé polynomials, remains valid for the seco…

Riemann-surface conjecture. For every $\boldsymbol f\in\mathcal A$, there exists an $(s+1)$-sheeted Riemann surface $\mathcal R=\mathcal R_{\boldsymbol f}$ such that the asymptotic…

Remainder-asymptotics conjecture. For every such domain $\mathcal D$,

Let $f$ belong to the Laguerre class $\mathscr L$, represented by … where $\alpha_j\in\mathbb R\setminus\mathbb Z$, $e_j\in\mathbb R$, and $e_1<\dots<e_{2q}$. Assume that every exp…

Let $f$ belong to the Laguerre class $\mathscr L$, represented by … where $\alpha_j\in\mathbb R\setminus\mathbb Z$, $e_j\in\mathbb R$, and $e_1<\dots<e_{2q}$. Assume that $1,f,f^2$…

Hermite–Padé approximants are constructed for a pair of Cauchy transforms, or Markov functions, associated with generally complex measures whose supports overlap. Their asymptotic…

Angelesco uniqueness conjecture. The measure $\mu=\lambda_1+\lambda_2$ is the only positive Borel measure in the plane satisfying these inequalities and $\mu(\mathbb C)=2$. This un…

Consider an Angelesco system with disjoint supports, and let $Q_{2n}$ be the Hermite–Padé denominator. Let $\vec\lambda=(\lambda_1,\lambda_2)$ be the equilibrium measure of the ext…

Let $Q_N$ be the relevant Hermite–Padé polynomial, with $n,N\to\infty$ and $N/n\to k>1$. For a subsequence, suppose … Let $\lambda(\mu)$ be the equilibrium measure associated with…

Let $Q_n$ be polynomials satisfying the orthogonality conditions from the setup, with weights depending on the polynomial through $\Phi_n(z;P)$. Assume the hypotheses of the stated…

Let ${\bf f}=(f_1,\ldots,f_d)$ and ${\bf m}=(m_1,\ldots,m_d)\in\mathbb{Z}_+^d\setminus\{{\bf 0}\}$, and suppose that the denominator $Q_{n,\bf m}$ is unique for all sufficiently la…

Nuttall's conjecture. The branch $h_0$ is the Cauchy transform of the limiting zero-counting measure:

The analytic theory of Hermite–Padé approximants studies diagonal approximants with multi-index $\vec{n}=(n,n,\ldots,n)$ to a vector of $p$ analytic functions. Nuttall's approach a…