In this paper, we are concerned with multivariate Gegenbauer approximation of functions defined in the $d$-dimensional hypercube. Two new and sharper bounds for the coefficients of multivariate Gegenbauer expansion of analytic functions are presented based on two different extensions of the Bernstein ellipse. We then establish an explicit error bound for the multivariate Gegenbauer approximation associated with an $\ell^q$ ball index set in the uniform norm. We also consider the multivariate approximation of functions with finite regularity and derive the associated error bound on the full grid in the uniform norm. As an application, we extend our arguments to obtain some new tight bounds for the coefficients of tensorized Legendre expansions in the context of polynomial approximation of parameterized PDEs.

中文翻译：

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超立方体中多元Gegenbauer近似的分析

在本文中，我们关注在 $d$ 维超立方体中定义的函数的多元 Gegenbauer 逼近。基于伯恩斯坦椭圆的两个不同扩展，给出了解析函数的多元 Gegenbauer 展开系数的两个新的更清晰的界限。然后，我们为与在统一范数中设置的 $\ell^q$ 球指数相关联的多元 Gegenbauer 近似建立显式误差界限。我们还考虑了具有有限规律性的函数的多元逼近，并在统一范数中导出完整网格上的相关误差界限。作为一个应用，我们扩展了我们的论点，以获得参数化偏微分方程多项式逼近上下文中张量勒让德展开系数的一些新的严格界限。