Analytic Elements in P-adic Analysis

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World Scientific, 1995 - Mathematics - 390 pages
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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.
 

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Contents

Absolute values and norms
1
Infraconnected sets
11
Monotonous and circular filters
18
Ultrametric absolute values and valuation functions vh fi on Kx
34
Ultrametric field extensions
39
Ultraproducts and spherically complete extensions
43
A study inCp the pnth roots of 1
51
The analytic elements
54
Tfilters and Tsequences
199
Examples and counterexamples about Tfilters
204
Characteristic property of the Tfilters
215
Applications of Tfilters
225
Integrally closed algebras HD
232
Absolute values on HD
243
Distinguished circular filters
248
Maximal ideals of infinite codimension
256

Composition of analytic elements
61
MultHDUD
67
Power series
70
Factorization of analytic elements
78
The MittagLeffler Theorem
82
Maximal ideals of codimension 1
89
Dual of a space HD
91
Algebras HD
95
Derivative of analytic elements
102
Valuation functions for analytic elements
109
Elements vanishing along a filter
114
Quasiminorated elements
120
Values and zeros of power series
124
Quasiinvertible elements
132
Zeros Theorem for power series
138
Image of a disk
145
Strictly injective analytic elements
148
Logarithm and exponential
155
A finite increasing property
159
Maximum principle
167
Analytic elements meromorphic in a hole
171
Motzkin factorization
175
Applications of the Motzkin factorization
190
Maximum in a circle with holes
194
Idempotent Tsequences
261
Tpolar sequences
269
Analytic extension through a Tfilter
272
Algebra
284
Meromorphic products
287
Collapsing meromorphic products
294
Injectivity MittagLeffier series and Motzkin products
298
Analytic functions and analytic elements
308
Infinite van der Monde matrices
316
padic analytic interpolation
324
Analytic elements with a zero derivative
328
Generalities on the differential equation y fy in HD
334
The differential equation y fy in algebras HD
338
The equation y fy in zero residue characteristic
341
The equation y fy in Cp with not quasiinvertible
346
The equation y fy in Cp with quasiinvertible
357
Residues and equation y fy
360
Equation g fg with gH HD
369
The padic Fourier transform
375
References
380
Definitions
385
Notations
388
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