双圆盘上Dirichlet型空间中Shanks猜想的多项式反例

中文摘要 AI 辅助

我们研究了双圆盘上Dirichlet型空间中的最优多项式逼近。给定非零函数f，其倒数的线性最优多项式逼近(opa)是仿射多项式p，使得pf在空间范数下最接近常数函数1。我们证明，对于Dirichlet型参数的每个正值，存在一个对称多项式f在闭双圆盘上没有零点，而相应的线性最优逼近在双圆盘内部有一个零点。这为这些空间上的弱Shanks猜想版本提供了多项式反例。该构造也推广到更高维的多圆盘和各向异性Dirichlet型空间。

英文摘要

We study optimal polynomial approximants in spaces of Dirichlet-type over the bidisc. Given a nonzero function f, the linear optimal polynomial approximant (opa) to its reciprocal is the affine polynomial p for which pf is closest to the constant function 1 in the space norm. We prove that, for every positive value of the Dirichlet-type parameter, there exists a symmetric polynomial f with no zeros on the closed bidisc, while the corresponding linear optimal approximant has a zero inside the bidisc. This provides polynomial counterexamples to the version of the Weak Shanks Conjecture on these spaces. The construction also extends to higher-dimensional polydiscs and to anisotropic Dirichlet-type spaces.

发表机构

• Universidad de La Laguna（拉帕鲁尼亚大学）

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