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. development at the uncomplicated items brought by means of L. Carlitz - resembling the Carlitz factorials, exponential and logarithm, and the orthonormal procedure of Carlitz polynomials - the writer develops one of those differential and crucial calculi.

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**Example text**

First we prove that the sequence {|cn |q n } is bounded. Assuming the opposite, we ﬁnd a strictly increasing sequence {nr } such that lim |cnr |q nr = ∞. Choosing, if necessary, an appropriate subsequence, we r→∞ may assume, for 0 ≤ n < nr , that |cn |q n < |cnr |q nr . 6) Suppose that u (0) = λ. 54) that t−1 fn (t) = gqn −1 (t) , Ln n = 0, 1, 2, . . 7) have the same roots t = xl , l < n, the same degrees and the same leading coeﬃcients). 7), nr lim ϕ(xnr ) = lim r→∞ r→∞ cn gqn −1 (xnr ) = λ. 14) of the Carlitz polynomials, we ﬁnd that gqn −1 (xn ) = Ln Dn n (−1)n−j j=0 n (qj −1)n x j n (−1)n−j = (−1)n + j=1 Ln j Dj Lqn−j where Ln j Dj Lqn−j = q −mn , x(q j −1)n 50 Chapter 2 mn = n + (q j − 1)n − qj − 1 − q j (n − j) = jq j − (1 + q + · · · + q j−1 ) > 0, q−1 so that gqn −1 (xn ) ≡ (−1)n (mod x).

The fact that the function t−1 γ(t) is continuous at t = 0 means that γ(t) is diﬀerentiable at t = 0. 2, |γn |q n → 0 for n → ∞. 11) with wn = L−1 n γn , so that wn → 0. Now we are in a position to prove the characterization result. 5 The function u = n=0 C0k+1 (O, K c ) if and only if k q nq |cn | → 0 for n → ∞. 13) In this case k sup |Dk u(t)| = sup q (n−k)q |cn |. 14) n≥k Proof. We use the identity fn−k , if n ≥ k, 0, if n < k. 15) n=k which implies the identity D u(t) k q −k ∞ −k cqn = n=k gqn−k −1 (t) , Ln−k t = 0.

Let u ∈ C0 (O, K c ), Dk u(t) = t−q ∆(k) u(t), k t ∈ O \ {0}. We will say that u ∈ C0k+1 (O, K c ) if Dk u can be extended to a continuous function on O. C0k+1 (O, K c ) can be considered as a Banach space over K c , with the norm sup |u(t)| + |Dk u(t)| . t∈O Note that coincides with the set of all diﬀerentiable Fq -linear functions O → K c . In this section we will obtain a characterization of functions from C01 (O, K c ) Calculus 49 C0k+1 (O, K c ) in terms of coeﬃcients of the expansion u = ∞ cn fn .

### Analysis in Positive Characteristic by Anatoly N. Kochubei

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