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 ℓ^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 parametrized PDEs.

中文翻译：

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超立方体中的多元Gegenbauer逼近分析

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Bernstein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和更清晰的界线。然后，我们建立约束与相关的多元盖根堡近似精确的错误ℓ ^q球指标设置在统一规范中。我们还考虑了具有有限规则性的函数的多元逼近，并得出了统一范数中完整网格上的相关误差范围。作为一种应用，我们扩展了参数以在参数化PDE的多项式逼近的情况下获得张量勒让德展开的系数的新的紧界。