The equioscillation theorem interleaves the Haar condition, the existence and uniqueness and strong uniqueness of the optimal Chebyshev approximation and its characterization by the equioscillation condition in a way that cannot extend to multivariate approximation: Rice~[\emph{Transaction of the AMS}, 1963] says ''A form of alternation is still present for functions of several variables. However, there is apparently no simple method of distinguishing between the alternation of a best approximation and the alternation of other approximating functions. This is due to the fact that there is no natural ordering of the critical points.'' In addition, in the context of multivariate approximation the Haar condition is typically not satisfied and strong uniqueness may hold or not. The present paper proposes an multivariate equioscillation theorem, which includes such a simple alternation condition on error extrema, existence and a sufficient condition for strong uniqueness. To this end, the relationship between the properties interleaved in the univariate equioscillation theorem is clarified: first, a weak Haar condition is proposed, which simply implies existence. Second, the equioscillation condition is shown to be equivalent to the optimality condition of convex optimization, hence characterizing optimality independently from uniqueness. It is reformulated as the synchronized oscillations between the error extrema and the components of a related Haar matrix kernel vector, in a way that applies to multivariate approximation. Third, an additional requirement on the involved Haar matrix and its kernel vector, called strong equioscillation, is proved to be sufficient for the strong uniqueness of the solution. These three disconnected conditions give rise to a multivariate equioscillation theorem, where existence, characterization by equioscillation and strong uniqueness are separated, without involving the too restrictive Haar condition. Remarkably, relying on optimality condition of convex optimization gives rise to a quite simple proof. Instances of multivariate problems with strongly unique, non-strong but unique and non-unique solutions are presented to illustrate the scope of the theorem.

翻译：等振荡定理通过Haar条件、最佳切比雪夫逼近的存在性与唯一性及强唯一性、以及等振荡条件的刻画相互交织，但其方式无法推广至多元逼近：Rice~[\emph{《美国数学会汇刊》}，1963]指出，“多变量函数仍存在某种形式的交错现象。然而，似乎没有简单方法区分最佳逼近的交错与其他逼近函数的交错。这是由于临界点缺乏自然排序所致。”此外，在多元逼近语境下，Haar条件通常不成立，而强唯一性可能成立也可能不成立。本文提出了一个多元等振荡定理，该定理包含关于误差极值的简单交错条件、存在性及强唯一性的充分条件。为此，首先澄清了单变量等振荡定理中交织性质之间的关系：首先提出弱Haar条件，该条件直接蕴含存在性；其次证明等振荡条件等价于凸优化的最优性条件，从而在不依赖唯一性的前提下刻画最优性；该条件被重新表述为误差极值与相关Haar矩阵核向量分量之间的同步振荡，使其适用于多元逼近；最后，对涉及的Haar矩阵及其核向量增加一个称为强等振荡的额外要求，并证明该条件是解强唯一性的充分条件。这三个相互独立的条件构成了多元等振荡定理，其中存在性、等振荡条件刻画及强唯一性被分离处理，避免了过于严格的Haar条件。值得注意，基于凸优化最优性条件可得出相当简洁的证明。本文还给出了具有强唯一解、非强唯一但唯一解以及非唯一解的多元问题实例，以阐明定理的适用范围。