In a recent paper, Br\"and\'en, Krasikov, and Shapiro consider root location preservation properties of finite difference operators. To this end, the authors describe a natural polynomial convolution operator and conjecture that it preserves root mesh properties. We prove this conjecture using two methods. The first develops a novel connection between the additive (Walsh) and multiplicative (Grace-Szeg\"o) convolutions, which can be generically used to transfer results from multiplicative to additive. We then use this to transfer an analogous result, due to Lamprecht, which demonstrates logarithmic root mesh preservation properties of a certain $q$-multiplicative convolution operator. The second method proves the result directly using a modification of Lamprecht's proof of the logarithmic root mesh result. We present his original argument in a streamlined fashion and then make the appropriate alterations to apply it to the additive case.

中文翻译：

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连接q-乘法卷积和有限差分卷积

在最近的一篇论文中，Br\"and\'en、Krasikov 和 Shapiro 考虑了有限差分算子的根位置保留特性。为此，作者描述了一个自然多项式卷积算子并推测它保留了根网格特性。我们证明这个猜想使用了两种方法。第一种方法在加法 (Walsh) 和乘法 (Grace-Szeg\"o) 卷积之间建立了一种新的联系，它通常可用于将结果从乘法转换为加法。然后我们使用它来传递一个类似的结果，由于 Lamprecht，它展示了某个 $q$-乘法卷积算子的对数根网格保留特性。第二种方法直接使用对数根网格结果的 Lamprecht 证明的修改来证明结果。