By Alain Escassut

The behaviour of the analytic components on an infraconnected set D in okay an algebraically closed whole ultrametric box is principally defined via the round filters and the monotonous filters on D, in particular the T-filters: zeros of the weather, Mittag-Leffler sequence, factorization, Motzkin factorization, greatest precept, injectivity, algebraic homes of the algebra of the analytic components on D, difficulties of analytic extension. this is often utilized to the differential equation y'=hy (y,h analytic components on D), analytic interpolation, p-adic crew duality on meromorphic items and to the p-adic Fourier remodel

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**Extra resources for Analytic elements in p-adic analysis**

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Now lei a to belong to a hole T of diameter p < R (resp. p < < R). 3 for Ler every r < R (resp. r > R), D has points a such that r < \a — a\ < R (resp. R < \a — a\ < r) and this ends the proof. Definition: Let J- be an increasing (resp. a decreasing) filter of center a and diameter R on D . [, (resp. r < R) , r ( a , r , R) (resp. r ( a , R, r)) contains some hole T m of D. A decreasing filter with no center T on D is said to be pierced if for every m £ IN, Dm \ Dm+i contains some hole Tm of D.

But then S is equal to xm and therefore P = xmT = \_] PjX^+Tn. j=o n But by hypothesis we assume P — xq. PjxJ*m an d therefore j=o \/3j\ < 1 whenever j < n. This contradicts the relation |/? 0 | = 1 and finishes the proof. Lemmas below will be useful in the sequel. 6: Let G be a subgroup of the multiplicative group (L*,-) included in C(0,1) and let u £ G. The bisection tp from G onto G defined as i/>(x) = ux is isometric. 7: Let (j,n) £ IN x IN* be such that j < n. 8: For every s £ IN, G3 is an Eisenstein Proof : First we suppose 5 = 1.

Hence we can define a surjective mapping g from / onto IN as g(i) = n whenever i G In- Now for every i G / , we put hi = big^. By (VV„), we have \\bn\\ < r o whenever n G IN, hence |/it|,- < ro for each i G / , and therefore (hi)i^j belongs to 71. We put h = (hi)iei, and UJ — ip(h). We will show (2) |a; — an\ < r n _ ! whenever n G IN*. Let n G IN* be fixed. It is seen that for every m > n, we have ||6 n — 6 m || < r n _ i , hence for every i G I we have | 6 i n — 6ijTn|i < r n _!. Beside, since ( / m ) m > n makes a partition of Xn_|_i, for every i G -Xn+i there exists m > n such that i G I m , and then we have |6;>n - h{\{ = \biin - bit9^\i = |6 i>n - 6ijTn|i < r n _ !