By Anatoly N. Kochubei

Dedicated to opposite numbers of classical buildings of mathematical research in research over neighborhood fields of optimistic attribute, this ebook treats confident attribute phenomena from an analytic perspective. development at the easy items brought by means of L. Carlitz - comparable to the Carlitz factorials, exponential and logarithm, and the orthonormal method of Carlitz polynomials - the writer develops a type of differential and critical calculi.

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**Example text**

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 . Consider functions hv : V → Fq of the form n−1 n−1 1 − (ψj (w) − aj )q−1 = hv (w) = 1 − (ψj (w) − ψj (v))q−1 . j=0 j=0 Since hv (w) = 1 for v = w and hv (w) = 0 when w = v (because at least one of the diﬀerences ψj (w) − ψj (v) is in this case a nonzero element of the ﬁnite ﬁeld Fq ), the Fq -span of all the hv is the whole set Maps(V, Fq ). Expanding the product deﬁning hv shows that hv is in the span of the Ψi since the exponents of the ψj in the product never exceed q − 1.

Given an orthonormal basis {ϕj }∞ 0 of the space C0 (O, K) of Fq -linear continuous functions, how do we construct an orthonormal basis {Φj }∞ 0 of the space C(O, K) of all continuous functions? How do we ﬁnd Orthonormal systems 23 the coeﬃcients of the expansions in such bases? While the second problem is connected with special properties of each basis, the ﬁrst one admits a general solution. Let us write every integer i ≥ 0 in the base q as i = α0 + α1 q + · · · + αn−1 q n−1 , 0 ≤ αj ≤ q − 1.

2, now u belongs to the Banach space H. 2, d λf˜n = λ1/q f˜n−1 , n ≥ 1, λ ∈ K c ; df˜0 = 0. It follows that ∞ dn u(t) = 1/q n ak k=n k−n tq , Dk−n so that 1/q n ak dn u(t) . 23) for the normalized Carlitz polynomials we ﬁnd that τ n dn = ∆(n) . 2). 2. Smooth functions. 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)| .