## The $$p$$-adic $$z$$-transform.(English)Zbl 0844.11074

Let $$a+ p^n \mathbb{Z}_p$$ be a ball in $$\mathbb{Z}_p$$. For a given $$z\in \mathbb{C}_p$$ let $${\mathfrak m}_z (a+p^n \mathbb{Z}_p)= z^a/ (1- z^{p^n})$$, where $$z\in \mathbb{C}_p$$, $$|z-1 |_p\geq 1$$. The $$p$$-adic distribution $${\mathfrak m}_z$$ defines a kind of $$z$$-transform by $$F(z)= \int_{\mathbb{Z}_p} f(x) {\mathfrak m}_z (x)$$. The author presents a systematic study of this transform. Moreover he gives some applications, e.g. to Mahler’s theorem on representations of continued functions $$f: \mathbb{Z}_p\to \mathbb{Q}_p$$ in the form $$f(x)= \sum^\infty_{n=0} a_n \binom x n$$, van der Put expansions and to the expansion of a $$p$$-adic function in a series of Sheffer polynomials.

### MSC:

 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.) 46S10 Functional analysis over fields other than $$\mathbb{R}$$ or $$\mathbb{C}$$ or the quaternions; non-Archimedean functional analysis
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### References:

 [1] Cassels, J.W.S. : Local Fields. Cambridge University Press, 1986. · Zbl 0595.12006 [2] Schikhof, W. : Ultrametric Calculus. Cambridge University Press, 1984. · Zbl 0553.26006 [3] Koblitz, N. : p-Adic Analysis : A Short Course on Recent Work. Cambridge University Press, 1980. · Zbl 0439.12011 [4] Amice, Y. - Fresnel, J. : Fonctions zêta p-adiques des corps de nombres abéliens réels. Acta Arithmetica, vol 20 (1972) p. 355-385. · Zbl 0217.04303 [5] Van Hamme, L. : Three generalizations of Mahler’s expansion for continuous functions on Zp. in “p-adic analysis” - vol 1454 (1990) p. 356-361, Springer Verlag. · Zbl 0716.39003 [6] Van Hamme, L. : Continuous operators which commute with translations on the space of continuous functions on Zp. in ”” J. Bayod, N. De Grande-De Kimpe, J. Martinez - Maurica p. 75-88, Marcel Dekker, New York (1992). · Zbl 0773.47039
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