整数集与实数集上复卡茨 - 伯恩斯坦泛函方程的连续解
Continuous solutions of the complex Kac--Bernstein functional equation on the integers and the real numbers
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中文总结 AI 辅助
研究整数集与实数集上复卡茨 - 伯恩斯坦泛函方程连续解的分类，利用\(\mathbb{Z}\)上分类推广一维环面\(\mathbb{T}\)上卡茨 - 伯恩斯坦定理，类似地推广\(\mathbb{R}\)上原定理，均从概率博雷尔测度推广到复博雷尔测度。
中文摘要 AI 辅助
本文对整数集\(\mathbb{Z}\)和实数集\(\mathbb{R}\)上复卡茨 - 伯恩斯坦泛函方程\(f_1 ( u + v ) f_2 ( u - v ) f_1 ( u' + v' ) f_2 ( u' - v' ) = f_1 ( u + v' ) f_2 ( u - v' ) f_1 ( u' + v ) f_2 ( u' - v )\)的连续解进行分类。利用\(\mathbb{Z}\)上的分类，将一维环面\(\mathbb{T}\)上由巴里什尼科夫 - 艾森伯格 - 斯塔德耶给出的卡茨 - 伯恩斯坦定理从概率博雷尔测度推广到复博雷尔测度。类似地，将\(\mathbb{R}\)上原卡茨 - 伯恩斯坦定理从概率博雷尔测度推广到复博雷尔测度。
英文摘要
In this paper, the continuous solutions of the complex Kac--Bernstein functional equation \[ f_1 ( u + v ) f_2 ( u - v ) f_1 ( u' + v' ) f_2 ( u' - v' ) = f_1 ( u + v' ) f_2 ( u - v' ) f_1 ( u' + v ) f_2 ( u' - v ) \] are classified for $ \mathbb{ Z } $ and $ \mathbb{ R } $. By using this classification for $ \mathbb{ Z } $, we consider a generalization of the Kac--Bernstein theorem on the one-dimensional torus $ \mathbb{ T } $ by Baryshnikov--Eisenberg--Stadje from probability Borel measures to complex Borel measures. Similarly, the original Kac--Bernstein theorem on $ \mathbb{ R } $ is generalized from probability Borel measures to complex Borel measures.
