This is probably the first book dedicated to this topic. The behaviour of the analytic elements on an infraconnected set D in K an algebraically closed complete ultrametric field is mainly explained by the circular filters and the monotonous filters on D, especially the T-filters: zeros of the elements, Mittag-Leffler series, factorization, Motzkin factorization, maximum principle, injectivity, algebraic properties of the algebra of the analytic elements on D, problems of analytic extension, factorization into meromorphic products and connections with Mittag-Leffler series. This is applied to the differential equation y'=hy (y,h analytic elements on D), analytic interpolation, injectivity, and to the p-adic Fourier transform.
Ultrametric absolute values and norms; infraconnected sets; monotonous and circular filters; the ultrametric absolute values on K(x); valuation functions u(h,mu) on K(x); hensel lemma; the analytic elements; factorization of analytic elements; the Mittage-Leffler theorem; derivative of analytic elements; elements vanishing along a monotonous filter; quasi-minorated elements; analytic elements meromorphic in a hole; Motzkin factorization; maximum in a circle with holes; T-filters and T-sequences; integrally closed algebras H(D); absolute values in algebras H(D); idempotent T-sequences; injectivity, Mittag-Leffler series and Motzkin products; generalities on the differential equation y'=fy in H(D); the equation y'=fy in zero residue characteristic; p-adic group duality; p-adic Fourier transform. (Part Contents).