Definition Mahler's theorem refers to a mathematical result established by Kurt Mahler concerning the approximation of p-adic functions by polynomials. Specifically, it provides a criterion for when a function defined on the p-adic integers can be uniformly approximated by polynomials, analogous to the classical Weierstrass approximation theorem in real analysis.
Overview Mahler's theorem is a foundational result in p-adic analysis, a branch of number theory dealing with p-adic numbers—completion of the rational numbers for a prime p, different from the real numbers. The theorem asserts that any continuous function from the p-adic integers ℤₚ to the p-adic numbers ℚₚ can be expressed as a uniformly convergent series of polynomial functions with coefficients tending to zero. This expansion is known as the Mahler expansion or Mahler series.
The result has significant applications in number theory, especially in the study of p-adic L-functions, modular forms, and interpolation of arithmetic functions.
Etymology/Origin The theorem is named after Kurt Mahler (1903–1988), a German-born mathematician who made substantial contributions to number theory, particularly in the fields of transcendental numbers and p-adic analysis. He introduced the theorem in a 1958 paper titled "An interpolation series for continuous functions of a p-adic variable," published in the Journal für die reine und angewandte Mathematik.
Characteristics
- Applies to continuous functions f: ℤₚ → ℚₚ.
- Represents such functions as infinite series:
f(x) = Σₙ₌₀^∞ aₙ·(x choose n),
where (x choose n) denotes the binomial coefficient generalized to p-adic arguments, and the coefficients aₙ are p-adic numbers satisfying limₙ→∞ aₙ = 0. - The convergence is uniform on ℤₚ.
- The coefficients aₙ can be explicitly computed using finite differences of the function f.
Related Topics
- p-adic analysis
- Weierstrass approximation theorem
- p-adic L-functions
- Continuous functions in number theory
- Binomial coefficient functions
- Kurt Mahler
- Non-Archimedean functional analysis