Mahler's theorem

In mathematics, Mahler's theorem, introduced by Kurt Mahler (1958), expresses any continuous p-adic function as an infinite series of certain special polynomials. This series is referred to as a Mahler series^[1] or Mahler expansion,^[2] or as Mahler's expansion.^[3] It is the p-adic counterpart to the Stone–Weierstrass theorem for continuous real-valued functions on a closed interval.

Statement

Let $(Δ f) (x) = f (x + 1) - f (x)$ be the forward difference operator. Then for any p-adic function $f : ℤ_{p} \to ℚ_{p}$ , Mahler's theorem states that $f$ is continuous if and only if its Newton series converges everywhere to $f$ , so that for all $x \in ℤ_{p}$ we have

f (x) = \sum_{n = 0}^{\infty} (Δ^{n} f) (0) (\binom{x}{n}),

where

(\binom{x}{n}) = \frac{x (x - 1) (x - 2) \dots (x - n + 1)}{n!}

is the $n$ th binomial coefficient polynomial. Here, the $n$ th forward difference is computed by the binomial transform, so that $(Δ^{n} f) (0) = \sum_{k = 0}^{n} (- 1)^{n - k} (\binom{n}{k}) f (k) .$ Moreover, we have that $f (x) := \sum_{n = 0}^{\infty} a_{n} (\binom{x}{n})$ is continuous if and only if the coefficients $a_{n} = (Δ^{n} f) (0) \to 0$ in $ℚ_{p}$ as $n \to \infty$ .

It is remarkable that as weak an assumption as continuity is enough in the p-adic setting to establish convergence of Newton series. By contrast, Newton series on the field of complex numbers are far more tightly constrained, and require Carlson's theorem to hold.

References

↑ Anashin, Vladimir (2021). "The p-adic Theory of Automata Functions". Advances in Non-Archimedean Analysis and Applications. STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health. pp. 9–113. doi:10.1007/978-3-030-81976-7_2. ISBN 978-3-030-81975-0.
↑ Duran, Ugur; Acikgoz, Mehmet (2019). "On applications for Mahler expansion associated with p-adic q-integrals". International Journal of Number Theory. 15: 67–84. doi:10.1142/S1793042118501737.
↑ Van Hamme, Lucien (1990). "Three generalizations of Mahler's expansion for continuous functions on ℤp". P-adic Analysis. Lecture Notes in Mathematics. Vol. 1454. pp. 356–361. doi:10.1007/BFb0091150. ISBN 978-3-540-53477-8.

Mahler, K. (1958), "An interpolation series for continuous functions of a p-adic variable", Journal für die reine und angewandte Mathematik, 1958 (199): 23–34, doi:10.1515/crll.1958.199.23, ISSN 0075-4102, MR 0095821, S2CID 199546556

[1] Anashin, Vladimir (2021). "The p-adic Theory of Automata Functions". Advances in Non-Archimedean Analysis and Applications. STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health. pp. 9–113. doi:10.1007/978-3-030-81976-7_2. ISBN 978-3-030-81975-0.

[2] Duran, Ugur; Acikgoz, Mehmet (2019). "On applications for Mahler expansion associated with p-adic q-integrals". International Journal of Number Theory. 15: 67–84. doi:10.1142/S1793042118501737.

[3] Van Hamme, Lucien (1990). "Three generalizations of Mahler's expansion for continuous functions on ℤp". P-adic Analysis. Lecture Notes in Mathematics. Vol. 1454. pp. 356–361. doi:10.1007/BFb0091150. ISBN 978-3-540-53477-8.

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