According to the famous and pioneering result of Laurent Schwartz, any closed translation invariant linear space of continuous functions on the reals is synthesizable from its exponential monomials. Due to a result of D. I. Gurevič there is no straightforward extension of this result to higher dimensions. Following Székelyhidi (Acta Math Hungar 153(1):120–142, 2017), with the aid of Gelfand pairs and K-spherical functions, K-synthesizability of K-varieties can be described. In this paper we contribute to this direction in the special case when K is the symmetric group of order d.

中文翻译：

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对称光谱合成

根据洛朗·施瓦茨（Laurent Schwartz）的著名和开创性结果，实数上连续函数的任何平移不变线性空间都可以从其指数单项式中合成。由于D. I.Gurevič的结果，无法将此结果直接扩展到更高的尺寸。继塞凯利希迪（Acta Math Hungar 153（1）：120–142，2017）之后，借助Gelfand对和K-球面函数，可以描述K-变体的K-可合成性。在本文中，当K为阶d的对称群时，在特殊情况下我们将为这一方向做出贡献。