By Anatoly N. Kochubei

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 perspective. development at the uncomplicated items brought by means of L. Carlitz - akin to the Carlitz factorials, exponential and logarithm, and the orthonormal process of Carlitz polynomials - the writer develops one of those differential and quintessential calculi.

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Dem Verlag und meinem Kollegen, Herrn Prof. P. C. Kendall, danke ich herzlich fer den Vorschlag, eine neue Ausgabe der "Vektoranalysis" in deutscher Ubersetzung vorzubereiten. Bei dieser Gelegenheit wurden einige kleine Fehler und andere Unebenheiten im Originaltext geandert. Eine weitere Verbesserung ist das hinzugefugte Kapitel uber kartesische Tensoren.

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**Example text**

Here the operators a ˜± (satisfying the same relation a ˜− a ˜+ −˜ a+ a ˜− = I) act on the Hilbert space of entire functions ∞ u(z) = zn cn √ , n! n=0 ∞ |cn |2 < ∞, z ∈ C, n=0 with the inner product (u1 , u2 ) = 1 π 2 u1 (z)u2 (z)e−|z| dz. C a− u)(z) = u (z). Instead of the Hermite functions, Here (˜ a+ u)(z) = zu(z), (˜ ∞ zn the orthonormal basis with the above properties is √ . n! n=0 An important object related to the CCR is the system of coherent states, generalized eigenfunctions (not necessarily belonging to the Hilbert space) of the annihilation operator.

Viewing the operator d as a kind of derivative, it is natural to introduce an appropriate antiderivative. Following the terminology used in the analysis over Zp (see [98]) we call it the indeﬁnite sum. Consider in C0 (O, K c ) the equation du = f, ∞ Suppose that f = f ∈ C0 (O, K c ). ϕk fk , ϕk ∈ K c . 18) ∞ ck fk and using k=0 the fact that ∞ 1/q ck fk−1 = k=1 ϕql , ∞ 1/q du = cl+1 fl l=0 we ﬁnd that cl+1 = l = 0, 1, 2, . . 18) uniquely up to the term c0 f0 (t) = c0 t, c0 = u(1). Fixing u(1) = 0 we obtain an Fq -linear bounded operator S on C0 (O, K c ), the operator of indeﬁnite sum: Sf = u.

Below it will be considered for z ∈ K c , |z| < q −1/(q−1) . 23 The Fq -linear functions eC and logC are inverse to each other. Proof. We note ﬁrst of all that eC and logC are mutually inverse as formal power series. Indeed, we have to prove that l (−1)l−n = n Dn Lql−n n=0 l (−1)n = n L Dq n=0 n l−n 0, for l > 0; 1, for l = 0, 0, for l > 0; 1, for l = 0. 15). 22. 61) and the ultra-metric inequality, n |logC (z)| ≤ sup q n |z|q . n≥0 The function ψz (s) = s|z| decreases for s > −(log |z|)−1 ; if |z| < q −1/(q−1) , q−1 n then ψz decreases for s > log q .