By Alain Escassut

ISBN-10: 9810222343

ISBN-13: 9789810222345

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

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**Sample text**

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 _ !

LI. Wee will denote te5 by , f/fr"'(a) (f0a(a ( a ) its by "(a) " ((a)) left side >int esp. ff'// (a)) >y )) (resp. (resp itive denvative right side (resp. left side ) derivative at a. ated to the For convenience, on L[x] we denote by v( . ) the valuationn associated the norm || . e. v(P) = v(P,0) whenever P G L[x]. F. F. For For every fi < v(a — b), w we have va(h,ii) — VJ,(/I,/X) . n n For For + For every even/ IR, 7V+(P,/i) IR,7V+ il ]] •. For every //i G IR R,JV (P,/ij + + (P, /i) — every // G IR, 7V IR, N~(P,fji) is equal to the number i=o of zeros admitted by P in the circle C(0,u?

Decreasing) distances holes sequence that runs an increasing (resp. decreasing) filter T will be just named an increasing (resp. decreasing) distances holes sequence and the filter T will be named the increasing (resp. decreasing) filter associated to the se quence (Tm^) i*1 ) 1 <,< a ( m ). If T has a center a , a will be named the center of the _ m6lN sequence (Tm^) i ,) will be called ~m€lN a decreasing distances holes sequence with no center. *

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