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.
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 Mahler's Theorem
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Source Mahler's theorem (Wikipedia)
Proved By
Source Mahler's theorem (Wikipedia)
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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)
Principal Result Of: Kurt Mahler, Lead sentence of the article on the result
In mathematics, Mahler's theorem, introduced by Kurt Mahler (1958), expresses any continuous p-adic function as an infinite series of certain special polynomials.
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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