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Jebel Quant Research（Jebel Quant Research）
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本文证明了Schmelzer和Trefethen关于φ-函数最优有理逼近收敛速率的猜想，即收敛速率由Halphen常数决定，并推广到更一般的函数类。
AI 中文摘要
函数 $\varphi_\ell(z)=(e^z-s_\ell(z))/z^\ell$，其中 $s_\ell$ 是 $e^z$ 的 $\ell-1$ 次泰勒多项式，是指数积分器所计算的函数，它们通过负实轴 $\mathbb{R}_-$ 上的有理逼近来计算这些函数。Schmelzer 和 Trefethen（2007/08）猜想，对于每个 $\ell$，这类最优逼近以经典“1/9”问题的速率收敛：$E_n(\varphi_\ell)^{1/n}\to H$，其中 $H$ 是 Halphen 常数。我们证明了这一猜想。其背后的定理给出了 $f=u_0+u_1\exp$ 的相同速率，其中 $u_0$ 为有理函数，$u_1\not\equiv0$ 为具有有限多个极点的亚纯函数，且 $\log|u_1(z)|=o(|z|)$，下界仅在远离 $\mathbb{R}_-$ 处需要，只要 $f$ 在 $\mathbb{R}_-$ 上没有极点（定理4.4）。对于有理函数 $u_1$，这是 Stahl 和 Schmelzer（2009）的定理1，其证明被推迟到一篇从未出现的手稿中；该假设允许例如 $u_1(z)=\sinh(\pi\sqrt z)/(\pi\sqrt z)$，其零点有无穷多个。证明是对 Gonchar 和 Rakhmanov（1989）的解读而非扩展。他们的定理涉及由带变权 $\Phi_n$ 的 Cauchy 型积分给出的序列，而 Schmelzer 和 Trefethen 的轮廓恒等式在去除主部后将 $u_1\exp$ 呈现为精确形式，其中 $\Phi_n(t)=u_1(-nt)e^{-nt}$。因子 $u_1(-nt)$ 在 $n$ 上呈次指数增长，而他们对 $\Phi_n$ 的假设除以 $2n$，因此它从外部场中消失：场、极值弧、$S$ 性质和常数都是他们的，未作改变。对于 $\varphi_\ell$，因子是 $t^{-\ell}$，其极点位于原点，这是逼近集的一个点，因此与每个允许的轮廓保持正距离。
英文摘要
The functions $φ_\ell(z)=(e^z-s_\ell(z))/z^\ell$, with $s_\ell$ the Taylor polynomial of $e^z$ of degree $\ell-1$, are what exponential integrators evaluate, and they evaluate them through rational approximations on the negative real axis $\mathbb{R}_-$. Schmelzer and Trefethen (2007/08) conjectured that the best such approximations converge at the rate of the classical `1/9'-problem: $E_n(φ_\ell)^{1/n}\to H$, Halphen's constant, for every $\ell$. We prove it. The theorem behind it gives the same rate for $f=u_0+u_1\exp$ with $u_0$ rational and $u_1\not\equiv0$ meromorphic with finitely many poles and $\log|u_1(z)|=o(|z|)$, the lower bound required only away from $\mathbb{R}_-$, whenever $f$ has no pole on $\mathbb{R}_-$ (Theorem~4.4). For rational $u_1$ this is Theorem 1 of Stahl and Schmelzer (2009), announced with its proof deferred to a manuscript that never appeared; the hypotheses admit, for instance, $u_1(z)=\sinh(π\sqrt z)/(π\sqrt z)$, whose zeros are infinite in number. The proof is a reading of Gonchar and Rakhmanov (1989) rather than an extension of them. Their theorem concerns a sequence given by a Cauchy-type integral with a varying weight $Φ_n$, and a contour identity of Schmelzer and Trefethen presents $u_1\exp$, once its principal parts are removed, in exactly that form, with $Φ_n(t)=u_1(-nt)e^{-nt}$. The factor $u_1(-nt)$ grows subexponentially in $n$, and their hypothesis on $Φ_n$ divides by $2n$, so it drops out of the external field: field, extremal arc, $S$-property and constant are all theirs, unchanged. For $φ_\ell$ the factor is $t^{-\ell}$, whose pole sits at the origin, a point of the approximation set and hence at positive distance from every admissible contour.
