Infinite Series Whose Topology of Convergence Varies From Point to Point

2023/07/01 by Maxwell C. Siegel, Siegel, Maxwell C.
Mathematics · #11R56 #12J10 #12J25 #26E30 #40A30 #43A70 #FOS: Mathematics #General Mathematics (math.GM) #Mathematical Dynamics and Fractals #Mathematical and Theoretical Analysis #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2307.00436

openalex publication_date 2023/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper catalogues a variety of examples concerning a type of function of a p-adic integer variable defined by a formal series expression we have dubbed "F-series". These series exhibit a new, previously undocumented form of point-wise convergence, one where the topology in which the limit of a sequence of functions \ f_n\_n≥1 converges depends on the point at which the sequence is evaluated. In a manner comparable to the adele ring of a number field, functions defined by F-series require considering different metric completions of an underlying field in order to be properly understood.

Related

• On the Disk of Convergence of Algebraic Power Series
• Pointwise Convergence of Fourier Series on the Ring of Integers of Local Fields with an Application to Gabor Systems
• Reprint: An interpolation series for continuous functions of a p-adic variable (1958)
• On p-Adic Power Series
• On convergence and rational summation of power series in p-adic field