2013/02/17 by J. M. Almira, Almira, J. M., Kh. F. Abu-Helaiel +1
Mathematics · #Mathematical and Theoretical Analysis #advanced mathematical theories #math.CA
paper · pdf · doi:10.48550/arxiv.1302.4086
12 pages, submitted to a journal
arxiv created 2013/02/17 · arxiv updated 2013/02/19
We prove a version of both Jacobi's and Montel's Theorems for the case of continuous functions defined over the field ℚp of p-adic numbers. In particular, we prove that, if Δh0m+1f(x)=0 for all x∈ℚp, and |h0|p=p-N0 then, for all x0∈ ℚp, the restriction of f over the set x0+pN0ℤp coincides with a polynomial px0(x)=a0(x0)+a1(x0)x+...+am(x0)xm. Motivated by this result, we compute the general solution of the functional equation with restrictions given by equation Δhm+1f(x)=0 (x∈ X and h∈ BX(r)=\x∈ X:‖x‖≤ r\), equation whenever f:X→ Y, X is an ultrametric normed space over a non-Archimedean valued field (\mathbbK,|...|) of characteristic zero, and Y is a ℚ-vector space. By obvious reasons, we call these functions uniformly locally polynomial.