In this work we blend interpolation theory with numerical integration, constructing an interpolator based on integrals over $n$-dimensional balls. We show that, under hypotheses on the radius of the $n$-balls, the problem can be treated as an interpolation problem both on a collection of $(n-1)$-spheres $ S^{n-1} $ and multivariate point sets, for which a wide literature is available. With the aim of exact quadrature and cubature formulae, we offer a neat strategy for the exact computation of the Vandermonde matrix of the problem and propose a meaningful Lebesgue constant. Problematic situations are evidenced and a charming aspect is enlightened: the majority of the theoretical results only deal with the centre of the domains of integration and are not really sensitive to their radius. We flank our theoretical results by a large amount of comprehensive numerical examples.

翻译：本文融合插值理论与数值积分方法，构造了基于$n$维球上积分的插值算子。研究表明，在$n$维球半径满足特定假设的条件下，该问题可转化为$（n-1）$维球面$ S^{n-1} $集合上的插值问题，以及多元点集上的插值问题，这两类问题已有丰富文献可资借鉴。为获得精确求积公式，本文提出了一套简洁策略用于精确计算问题的范德蒙矩阵，并定义了具有明确意义的勒贝格常数。文中揭示了若干困难情形，同时阐明了一个引人注目的特性：大多数理论结果仅与积分域中心相关，而对半径变化不敏感。我们通过大量详尽的数值实验对理论结果进行了验证。