Escassut A. Analytic Eelements in p-adic Analysis

World Scientific Publishing Company, 1995. — x, 390 p. — ISBN 981-02-2234-3.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.Absolute values and norms
lnfraconnected sets
Monotonous and circular filters
Ultrametric absolute values and valuation functions v(h, µ) on K(x)
Hensel Lemma
Ultrametric field extensions
Ultraproducts and spherically complete extensions
A study in C_p , the pⁿ-th roots of 1
Algebras R(D)
The analytic elements
Composition of analytic elements
Mult(H(D), U_D)
Power series
Factorization of analytic elements
The Mittag-Leffler Theorem
Maximal ideals of codimension 1
Dual of a space H(D)
Algebras H(D)
Derivative of analytic elements
Valuation functions for analytic elements
Elements vanishing along a monotonous filter
Quasi-minorated elements
Values and zeros of power series
Quasi-invertible elements
Zeros theorem for power series
Image of a disk
Strictly injective analytic elements
Logarithm and exponential
A finite increasing property
Maximum principle
Analytic elements meromorphic in a hole
Motzkin factorization
Applications of the Motzkin factorization
Maximum in a circle with holes
T-filters and T-sequences
Examples and counter-examples about T-filters
Characteristic property of the T-filters
Applications of T-filters
Integrally closed algebras H(D)
Absolute values on H(D)
Distinguished circular filters
Maximal ideals of infinite codimension
Idempotent T-sequences
T-polar sequences
Analytic extension through a T-filter
Algebra H(D)/I₀(F)
Meromorphic products
Collapsing meromorphic products
lnjectivity, Mittag-Leffier series and Motzkin products
Analytic functions and analytic elements
Infinite van der Monde matrices
p-adic analytic interpolation
Analytic elements with a zero derivative
Generalities on the differential equation y' = fy in H(D)
The differential equation y' = fy in algebras H(D)
The equation y' = fy in zero residue characteristic
The equation y' = fy in C_p with f not quasi-invertible
The equation y' = fy in C_p with f quasi-invertible
Residues and equation y' = fy
Equation g' = fg with gⁿ є H(D)
The p-adic Fourier transform
References
Definitions
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