The problem of verifying the nonnegativity of a function on a finite abelian group is a long-standing challenging problem. The basic representation theory of finite groups indicates that a function on a finite abelian group can be written as a linear combination of characters of irreducible representations of by , where is the dual group of consisting of all characters of and is the of at . In this paper, we show that by performing the fast (inverse) Fourier transform, we are able to compute a sparse Fourier sum of squares (FSOS) certificate of on a finite abelian group with complexity that is quasi-linear in the order of and polynomial in the FSOS sparsity of . Moreover, for a nonnegative function on a finite abelian group and a subset , we give a lower bound of the constant such that admits an FSOS supported on . We demonstrate the efficiency of the proposed algorithm by numerical experiments on various abelian groups of orders up to 10. As applications, we also solve some combinatorial optimization problems and the sum of Hermitian squares (SOHS) problem by sparse FSOS.

中文翻译：

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计算拟线性时间内有限阿贝尔群上的稀疏傅里叶平方和


验证有限交换群上函数的非负性问题是一个长期存在的挑战性问题。有限群的基本表示论表明，有限阿贝尔群上的函数可以写成 by 的不可约表示的特征的线性组合，其中 是由 的所有特征组成的对偶群，是 at 的特征。在本文中，我们表明，通过执行快速（逆）傅里叶变换，我们能够计算有限交换群上的稀疏傅里叶平方和（FSOS）证书，其复杂度为准线性，其顺序为 和FSOS 稀疏性中的多项式 .此外，对于有限阿贝尔群和子集 上的非负函数，我们给出常数的下界，使得允许 FSOS 支持。我们通过对高达 10 阶的各种阿贝尔群的数值实验证明了所提出算法的效率。作为应用，我们还通过稀疏 FSOS 解决了一些组合优化问题和埃尔米特平方和 (SOHS) 问题。