Analytic Elements in p-ADIC Analysis

By: Alain Escassut

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Hardcover | 1 December 1994

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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. This is applied to the differential equation y'=hy (y,h analytic elements on D), analytic interpolation, p-adic group duality on meromorphic products and to the p-adic Fourier transform.

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Introduction
Absolute values and norms	p. 1
Infraconnected sets	p. 11
Monotonous and circular filters	p. 18
Ultrametric absolute values and valuation functions v(h, [mu]) on K(x)	p. 27
Hensel Lemma	p. 34
Ultrametric field extensions	p. 39
Ultraproducts and spherically complete extensions	p. 43
A study in C[subsript p], the p[superscript n]-th roots of 1	p. 47
Algebras R(D)	p. 51
The analytic elements	p. 54
Composition of analytic elements	p. 61
Mult(H(D),U[subscript D])	p. 67
Power series	p. 70
Factorization of analytic elements	p. 78
The Mittag-Leffler Theorem	p. 82
Maximal ideals of codimension 1	p. 89
Dual of a space H(D)	p. 91
Algebras H(D)	p. 95
Derivative of analytic elements	p. 102
Valuation functions for analytic elements	p. 109
Elements vanishing along a filter	p. 114
Quasi-minorated elements	p. 120
Values and zeros of power series	p. 124
Quasi-invertible elements	p. 132
Zeros Theorem for power series	p. 138
Image of a disk	p. 145
Strictly injective analytic elements	p. 148
Logarithm and exponential	p. 155
A finite increasing property	p. 159
Maximum principle	p. 167
Analytic eleniets meromorphic in a hole	p. 171
Motzkin factorization	p. 175
Applications of the Motzkin factorization	p. 190
Maximum in a circle with holes	p. 194
T-filters and T-sequences	p. 199
Examples and counter-examples about T-filters	p. 204
Characteristic property of the T-filters	p. 215
Applications of T-filters	p. 225
Integrally closed algebras H(D)	p. 232
Absolute values on H(D)	p. 243
Distinguished circular filters	p. 248
Maximal ideals of infinite codimension	p. 256
Idemoptent T-sequences	p. 261
T-polar sequences	p. 269
Analytic extension through a T-filter	p. 272
Algebra [actual symbol not reproducible]	p. 284
Meromorphic products	p. 287
Collapsing meromorphic products	p. 294
Injectivity, Mittag-Leffler series and Motzkin products	p. 298
Analytic functions and analytic elements	p. 308
Infinite van der Monde matrices	p. 316
p-adic analytic interpolation	p. 324
Analytic elements with a zero derivative	p. 328
Generalities on the differential equation y' = fy in H(D)	p. 334
The differential equation y' = fy in algebras H(D)	p. 338
The equation y' = fy in zero residue characteristic	p. 341
The equation y' = fy in C[subscript p] with f not quasi-invertible	p. 346
The equation y' = fy in C[subscript p] with f quasi-invertible	p. 352
Residues and equation y' = fy	p. 360
Equation g' = fg with [actual symbol not reproducible]	p. 369
The p-adic Fourier transform	p. 375
References	p. 380
Definitions	p. 385
Notations	p. 388
Table of Contents provided by Blackwell. All Rights Reserved.

Analytic Elements in p-ADIC Analysis

At a Glance

Hardcover

Shipping

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