In this paper we prove that if a multivariate function of a certain smoothness class is represented by a sum of $k$ arbitrarily behaved ridge functions, then it can be represented by a sum of $k$ ridge functions of the same smoothness class and a polynomial of degree at most $k-1$. This solves the problem posed by A. Pinkus in his monograph “Ridge Functions” up to a multivariate polynomial.

中文翻译：

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岭函数的平滑和的表示问题

在本文中，我们证明如果某个平滑度类的多元函数由的和表示 $ķ$ 任意表现的岭函数，则可以用 $ķ$ 相同平滑度等级的岭函数和度的多项式 $ķ-\mathrm{1个}$。这解决了A. Pinkus在其专着“ Ridge Functions”中提出的问题，直至多元多项式。