By Anatoly N. Kochubei

Dedicated to opposite numbers of classical buildings of mathematical research in research over neighborhood fields of confident attribute, this publication treats confident attribute phenomena from an analytic perspective. development at the uncomplicated gadgets brought by means of L. Carlitz - similar to the Carlitz factorials, exponential and logarithm, and the orthonormal process of Carlitz polynomials - the writer develops a type of differential and vital calculi.

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19, for j < δn the reduction fj is a well-defined mapping from Fq [x]/(π n ) to Fq [x]/(π) ∼ = Fπ . Here (π n ) and (π) are the principal ideals in Fq [x] generated by π n and π, respectively. For 0 ≤ j, k ≤ δn − 1, the δn × δn matrix fj (xk ) is triangular, with all diagonal entries equal to 1. Since 1, x, . . , xδn−1 is an Fq -basis of Orthonormal systems 35 ∼ Fq [x]/(π n ), it follows that the δn reduced functions fj form Oπ /(π n ) = an Fπ -basis of HomFq (Fπ [[π]]/(π n ), Fπ ). 5 we come to the conclusion that {fj } is an orthonormal basis in C0 (Oπ , Kπ ).

13) was proved by Conrad [29] who also found its analog for the case of characteristic 0. Apart from the above explicit constructions of bases for the positive characteristic case, there are also various general results, applicable both for the cases of our field K, the field Qp of p-adic numbers, and its finite extensions. See [3, 7, 15, 19, 20, 29, 104, 107, 118]. Further constructions of orthonormal bases in C0 (O, K) are provided by an appropriate version of the umbral calculus; see Chapter 2 below.

41) was first proposed by Carlitz [23] for the case of the Carlitz polynomials. 17 proved in [62]. 45) is taken from the dissertation by Yang [123]. It was noticed by Jeong [53] that all the constructions are applicable also to the hyperdifferentiations. 13) was proved by Conrad [29] who also found its analog for the case of characteristic 0. Apart from the above explicit constructions of bases for the positive characteristic case, there are also various general results, applicable both for the cases of our field K, the field Qp of p-adic numbers, and its finite extensions.

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