2022/07/21 by Mateusz Krukowski, Krukowski, Mateusz
Mathematics · #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.2207.10706
openalex publication_date 2022/07/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The primary motivation behind this paper is an attempt to provide a thorough explanation of how the Mellin transform arises naturally in a process akin to the construction of the celebrated Gelfand transform. We commence with a study of a class of Schwartz functions S(ℝ+), where ℝ+ is the set of all positive real numbers. Various properties of this Fréchet space are established and what follows is an introduction of the Mellin convolution operator, which turns S(ℝ+) into a commutative Fréchet algebra. We provide a simple proof of Mellin-Young convolution inequality and go on to prove that the structure space Δ(S(ℝ+),⋆) (the space of nonzero, linear, continuous and multiplicative functionals m:S(ℝ+)\longrightarrow ℝ) is homeomorphic to ℝ. Finally, we show that the Mellin transform arises in a process which bears a striking resemblance to the construction of the Gelfand transform.