By Anatoly N. Kochubei

ISBN-10: 0511517777

ISBN-13: 9780511517778

ISBN-10: 0521509777

ISBN-13: 9780521509770

Dedicated to opposite numbers of classical constructions of mathematical research in research over neighborhood fields of confident attribute, this booklet treats confident attribute phenomena from an analytic standpoint. development at the simple items brought via L. Carlitz - comparable to the Carlitz factorials, exponential and logarithm, and the orthonormal procedure of Carlitz polynomials - the writer develops one of those differential and necessary calculi.

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Note that ϕj = Φqj . 41) is called the extension of the basis {ϕj } by digit expansions. 13 (“Digit principle”) The extension of an orthonormal basis of C0 (O, K) by digit expansions produces an orthonormal basis for C(O, K). Proof. 5 we reduce our problem to that of linear algebra over the ﬁnite ﬁeld Fq . The reduction of the space C0 (O, K) is HomFq (O, Fq ) = Fq ϕj where the reductions {ϕj }∞ 0 form an Fq -basis j≥0 of HomFq (O, Fq ). Here HomFq (O, Fq ) denotes the set of all continuous Fq linear maps (with respect to the discrete topology in Fq ).

Extend the functionals ψj to a system 24 Chapter 1 of polynomial functions on V by using digit expansions, that is for 0 ≤ i ≤ q n − 1 write i = α0 + α1 q + · · · + αn−1 q n−1 and set α n−1 Ψi = ψ0α0 · · · ψn−1 , so that ψj = Ψqj and Ψ0 = 1. We need to show that the functions Ψi form a basis of Maps(V, Fq ). By a dimension count, it suﬃces to prove that the Ψi span Maps(V, Fq ). Let {v0 , . . , vn−1 } be the dual basis to {ψ0 , . . , ψn−1 }. For v ∈ V , write v = a0 v0 + a1 v1 + · · · + an−1 vn−1 where aj ∈ Fq .

75) a+ and a− act upon the basis as follows: a+ fi−1 = [i]fi , a− fi = fi−1 , i ≥ 1; a− f0 = 0. 78). 78) |c| < q − q−1 , then q u(t) = λ− q−1 eC (tz), z = logC (c). 79). Proof. 21) of the Carlitz polynomials. 77). We can write ∞ cn fn (t), u(t) = t ∈ O, cn → 0. n=0 Applying the operator d, we get that ∞ c1/q n fn−1 , λu = n=1 and, by the uniqueness of the expansion, that 1/q cn+1 = λcn , n = 0, 1, . . , whence cn = λq n n +q n−1 +···+q q c0 = µ−1 (c0 µ)q , n n = 1, 2, . . , q where µ = λ q−1 .

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

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