37 problems

Let $P_{[p/q]}^{(c)}(x)$ denote the $[p/q]$ Padé approximant associated with the square-root expansion at $c$, and let $\beta_m^2(x,\sqrt{c})$ be the square biroot approximation. S…

Let $f\in H(E)$, and let $\rho_n(f,E)$ be the error of best rational approximation on $E$, with asymptotic rate parameter $\rho(f,E)$. Gonchar's liminf conjecture. For every such f…

Karpenkov's conjecture. For every real $\xi$, there are infinitely many integer pairs $(a,b)$ such that

Conjecture on cubic-order $a/b^2$-approximations. There exists a constant $c(\alpha)$ such that

Let $\epsilon>0$ be any small positive real number and let $N\geq 1$. Suppose that $\alpha$ has a rational approximation $a/q$ with integers $a,q$, $1\leq q\leq N^2$, $(a,q)=1$, an…

Let $\alpha$ be a real number, let $N\geq 1$ be real, and let $0\leq\beta\leq 1$. The integers $a_1,a_2$ and positive integers $q_1,q_2$ satisfy $1\leq q_1,q_2\leq N$. Interpolated…

Higher-power modulus spacing conjecture. The local multiplicity of $S_{k,Q}$ at scale $Q^{-k-1}$ should satisfy

Spacing conjecture for square moduli.

Let $\alpha>0$ and $\beta\in[0,2)$, and define the V-shaped domain … Let $r_N(z)$ be a lightning-plus-polynomial approximation to $z^\alpha$ with tapered lightning poles … and a po…

Let $\alpha>0$, let $N_1$ be the number of tapered lightning poles … and let $r_N(x)$ be the corresponding lightning-plus-polynomial approximation, with coefficients…

Let $n$ be the number of poles, and consider best uniform real-pole rational approximants to $\exp(-z)$ at the fixed time point $t=1$. Write the poles as $\sigma_1,\dots,\sigma_n$,…

Kruyswijk–Meijer conjecture. As $N\to\infty$,

Let $p\in\mathbb{N}$, let $\operatorname{RePU}_p(x)=\max(x,0)^p$, and let $\tilde{R}_n(x)=\frac{1}{2}(x^p+x^pr_n(x))$, where $r_n$ is as in the stated rational approximation constr…

For a positive expansion parameter $c$, let $\beta_m^2(1,c)$ denote the factored square biroot evaluated at $1$. Factored square-biroot identity. The value is … The source presents…

Generalized Binomial Biroot Conjecture. The approximations converge according to

Let $\widetilde p$ and $\widetilde q$ be the numerator and denominator in a barycentric rational approximant $\widetilde\xi=\widetilde p/\widetilde q$, let $\ell$ denote the number…

Let $0<\alpha<1$, let $N=N_1+N_2$, and let … where $\sigma>0$ and $C>0$. Define the lightning plus polynomial approximant … where $b_{N_2}$ is a polynomial and…

Herremans, Huybrechs and Trefethen's conjecture. There exist coefficients $\{a_j\}_{j=1}^{N_1}$ and a polynomial $P_{N_2}$ with $N_2=\mathcal{O}(\sqrt{N_1})$ such that

Let the true leading-order behaviour have a singularity at $x=x_{\mathrm{s}}$, and let the nearest pole to this point in a rational approximation be at $x=p_1$. Pole-location accur…

Let $-1\leq\beta<\alpha<0$, and let $E_{m,m}(x^\alpha,(0,1];x^\beta)$ denote the rational approximation error in the notation of the paper. Negative-exponent approximation conjectu…

Let $\beta$ and $\alpha$ be parameters in the setting of Theorem $\text{alpha-beta-subopt}$, and let $C_{\alpha,\beta}$ and $C_{1-\alpha,1-\beta}$ denote constants independent of…

Let $a_1<0<b_2$ and consider the best rational approximation of $\sqrt{\lambda}$ on the indefinite interval $[a_1,b_2]$. Indefinite-interval approximation conjecture. As the ration…

Let $K_1$ and $K_2$ be compact subsets of the plane, and let $\alpha$ denote continuous analytic capacity. Vitushkin's quasiadditivity conjecture. There exists a constant $C$ such…

Let $f(t)$ be an exponential sum and let ESPIRA-II, MPM, and ESPRIT denote the corresponding parameter-estimation methods applied to equidistant, possibly noisy samples of $f$. Sta…

Let $\vert x\vert$ be approximated on $[-1,1]$ by real polynomials of degree at most $n$, and let $\mathcal{E}_n(\vert x\vert;[-1,1])$ denote the corresponding best uniform approxi…