By Anatoly N. Kochubei

ISBN-10: 0511517777

ISBN-13: 9780511517778

ISBN-10: 0521509777

ISBN-13: 9780521509770

Dedicated to opposite numbers of classical buildings of mathematical research in research over neighborhood fields of confident attribute, this booklet treats confident attribute phenomena from an analytic perspective. construction at the simple gadgets brought through L. Carlitz - reminiscent of the Carlitz factorials, exponential and logarithm, and the orthonormal process of Carlitz polynomials - the writer develops a type of differential and essential calculi.

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It is interesting that, in contrast to the classical case, all the above p-adic operators a± are bounded. The problem of existence of linear bounded representations of CCR over valued fields of arbitrary rank is investigated in [56]. Below we construct a characteristic p representation of CCR. Just as in the above characteristic zero cases, the basic objects happen to be related to the main special functions of the corresponding branch of analysis – this time, to the Carlitz polynomials and the Carlitz exponential.

72) we have in this case a− Pn = Pn−1 , n ≥ 1; a+ Pn = (n + 1)Pn+1 , a− P0 = 0, n ≥ 0, 42 Chapter 1 so that (a+ a− )Pn = nPn and, as before, a− a+ − a+ a− = I. Here {Pn } is the Mahler basis of C(Zp , Qp ) (see [98]), that is Pn (t) = t(t − 1) · · · (t − n + 1) , n ≥ 1; n! P0 (t) ≡ 1. The analogs of coherent states are the functions fλ (t) = (1 + λ)t , where |λ|p < 1. Various p-adic analogs of the Bargmann–Fock representation are given in [2, 57, 60]. It is interesting that, in contrast to the classical case, all the above p-adic operators a± are bounded.

Below it will be considered for z ∈ K c , |z| < q −1/(q−1) . 23 The Fq -linear functions eC and logC are inverse to each other. Proof. We note first of all that eC and logC are mutually inverse as formal power series. Indeed, we have to prove that l (−1)l−n = n Dn Lql−n n=0 l (−1)n = n L Dq n=0 n l−n 0, for l > 0; 1, for l = 0, 0, for l > 0; 1, for l = 0. 15). 22. 61) and the ultra-metric inequality, n |logC (z)| ≤ sup q n |z|q . n≥0 The function ψz (s) = s|z| decreases for s > −(log |z|)−1 ; if |z| < q −1/(q−1) , q−1 n then ψz decreases for s > log q .

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Analysis in Positive Characteristic by Anatoly N. Kochubei

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