Download Analysis in Positive Characteristic by Anatoly N. Kochubei PDF

By Anatoly N. Kochubei

Dedicated to opposite numbers of classical constructions of mathematical research in research over neighborhood fields of confident attribute, this e-book treats confident attribute phenomena from an analytic point of view. development at the uncomplicated items brought through L. Carlitz - similar to the Carlitz factorials, exponential and logarithm, and the orthonormal method of Carlitz polynomials - the writer develops one of those differential and vital calculi.

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80) an u(t) = Dn n=0 44 Chapter 1 The norm in H is given by u H = sup |an |. 80) defines a holomorphic function for n tq |t| < q −1/(q−1) . It is obvious that the sequence of functions f˜n (t) = , Dn n = 0, 1, 2, . , is an orthonormal basis of H. The desired representation is given by the following operators on the space H: a ˜+ = τ, a ˜− = d. Note that the form of the operators a ˜± is only slightly different from that ± of a , but they act on a different Banach space. 76) hold for the operators a ˜± , with f˜n substituted for fn .

43) j+l=i i gj (t)Gl (s). 44) (iii) If 0 ≤ l < q ν , k ≥ 0, ν ∈ N, then Gk (m)gl (m) = m∈Fq [x] deg m<ν 0, if k + l = q ν − 1, (−1)ν , if k + l = q ν − 1. 0, if k + l = q ν − 1, (−1)ν , if k + l = q ν − 1. 46) Proof. 42) follows from the congruence n−1 n−1 αj ≡ j=0 (mod q − 1). αj q j j=0 Similarly we get the second equality if αj < q−1 for all j. If some αj = q−1, then for that j we get gαj qj (ξt) = gαj qj (t), if ξ = 0. Therefore we come to the required equality assuming that ξ = 0. 43), note that αj αj fj (t + s) αj = (fj (t) + fj (s)) = l=0 αj fj (t)l fj (s)αj −l , l 26 Chapter 1 whence n−1 αj Gj (t + s) = j=0 lj =0 αj fj (t)lj fj (s)αj −lj lj αn−1 α0 ··· = l0 =0 ln−1 =0 αn−1 α0 ··· f0 (t)l0 · · · fn−1 (t)ln−1 l0 ln−1 × f0 (s)α0 −l0 · · · fn−1 (s)αn−1 −ln−1 αn−1 α0 ··· = l0 =0 ln−1 =0 α0 αn−1 Gβ (t)Gi−β (s), ··· l0 ln−1 with β = l0 + l1 q + · · · + ln−1 q n−1 .

15) of the additive Carlitz polynomials were proved by Carlitz in the seminal paper [22] where the polynomials were introduced for the first time. Our proofs follow [45]. 16) is due to Wagner [119]. 8) were first studied by Wagner [118, 119]. There are several different proofs of this result [118, 119, 43, 61, 29, 32]; we followed [119]. 10 was proved in [64]. Hyperdifferentiations were introduced by Hasse [48] and studied in a more general context in [108, 49, 99]; for various generalizations see [112] and references therein.

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