Continuous solutions of the complex Kac--Bernstein functional equation on the integers and the real numbers

2026/07/27 by Takashi Satomi
#math.CA #math.CO #math.GR #math.PR

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Abstract

In this paper, the continuous solutions of the complex Kac--Bernstein functional equation f₁ ( u + v ) f₂ ( u - v ) f₁ ( u' + v' ) f₂ ( u' - v' ) = f₁ ( u + v' ) f₂ ( u - v' ) f₁ ( u' + v ) f₂ ( u' - v ) are classified for ℤ and ℝ . By using this classification for 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 ℝ is generalized from probability Borel measures to complex Borel measures.

Citations

• On the Skitovich-Darmois theorem for some locally compact Abelian groups
• Independent random variables on Abelian groups with independent the sum and difference
• The Skitovich-Darmois theorem for discrete and compact totally disconnected Abelian groups
• The Skitovich-Darmois theorem for finite Abelian groups

Related

• Solution of the Kac--Bernstein functional equation on Abelian groups in the class of positive functions
• Bernstein-Markov measures and Toeplitz theory
• Loewner Theory for Bernstein functions I: evolution families and differential equations
• A Complex Borel-Bernstein Theorem
• On Bernstein-type theorems