The equioscillation condition is extended to multivariate approximation. To this end, it is reformulated as the synchronized oscillations between the error maximizers and the components of a related Haar matrix kernel vector. This new condition gives rise to a multivariate equioscillation theorem where the Haar condition is not assumed and hence the existence and the characterization by equioscillation become independent of uniqueness. This allows the theorem to be applicable to problems with no strong uniqueness or even no uniqueness. A technical additional requirement on the involved Haar matrix and its kernel vector is proved to be sufficient for strong uniqueness. Instances of multivariate problems with strongly unique, unique and nonunique solutions are presented to illustrate the scope of the theorem.

翻译：等振荡条件被推广至多元逼近。为此，将其重新表述为误差最大化点与相关哈尔矩阵核向量分量之间的同步振荡。这一新条件导出了无需假设哈尔条件的多元等振荡定理，从而使得等振荡的存在性与刻画特性独立于唯一性。这使得该定理适用于无强唯一性甚至无唯一性的问题。进一步证明，对涉及的哈尔矩阵及其核向量施加一项技术性附加要求足以保证强唯一性。通过给出具有强唯一解、唯一解及非唯一解的多元问题实例，阐明了该定理的适用范围。