Hirschman and Widder introduced a class of Pólya frequency functions given by linear combinations of one-sided exponential functions. The members of this class are probability densities, and the class is closed under convolution but not under pointwise multiplication. We show that, generically, a polynomial function of such a density is a Pólya frequency function only if the polynomial is a homothety, and also identify a subclass for which each positive-integer power is a Pólya frequency function. We further demonstrate connections between the Maclaurin coefficients, the moments of these densities, and the recovery of the density from finitely many moments, via Schur polynomials.

Hirschman 和 Widder 介绍了一类由单边指数函数的线性组合给出的 Pólya 频率函数。该类的成员是概率密度，并且该类在卷积下是封闭的，但在逐点乘法下不是封闭的。我们证明，一般来说，只有当多项式是同位时，这种密度的多项式函数才是 Pólya 频率函数，并且还确定了每个正整数幂都是 Pólya 频率函数的子类。我们通过 Schur 多项式进一步证明了 Maclaurin 系数、这些密度的矩以及从有限多个矩中恢复密度之间的联系。