Mahler's theorem, introduced by Kurt Mahler in 1958, is the p-adic counterpart to the Stone-Weierstrass theorem. It states that a function from the p-adic integers to the p-adic rationals is continuous if and only if it can be written as a convergent infinite series of binomial-coefficient polynomials built from the forward difference operator, its Newton series. The theorem needs only the weak hypothesis of continuity to guarantee this convergence in the p-adic setting, a striking contrast with the real or complex case, where a comparable result requires the much stronger hypotheses of Carlson's theorem. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementA function from the p-adic integers to the p-adic rationals is continuous if and only if its Newton series converges everywhere to it. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Proved By
Source Mahler's theorem (Wikipedia)
Sources
1. Mahler's theorem (Wikipedia)
Statement
is continuous if and only if its Newton series converges everywhere to
Introduction, first sentence
introduced by Kurt Mahler (1958)
Proved By: Kurt Mahler, Lead paragraph
In mathematics, Mahler's theorem, introduced by Kurt Mahler (1958), expresses any continuous p-adic function as an infinite series of certain special
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