## Analytic Elements in P-adic AnalysisThis 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 |

380 | |

Definitions | 385 |

Notations | 388 |

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### Common terms and phrases

absolute value admits algebraically closed analytic elements annulus belongs to H(D bijection Cauchy filter Chapter clearly codimension converges in H(D Corollary decreasing filter defined definition denote diam(D diameter distances holes sequence distinguished circular filter easily seen element of H(D ends the proof equal f belongs filter of center finishes proving finitely H(D U T Hb(D homographic function homomorphism idempotent identically zero increasing resp increasing T-filter infraconnected set integers invertible in H(D isomorphic Let f Let f(x Let h Let q limsup logr maximal ideal meromorphic product Mittag-Leffler series monotonous filter Motzkin factor norm Notations obtain obviously assume open set p-adic analysis pierced filter polar sequence power series proof of Theorem Proposition quasi-invertible quasi-minorated radius of convergence rational functions residue characteristic secant semi-norm sequence an)neiN strictly injective strictly vanishing strongly copiercing sequence suppose surjective ultrametric uniform convergence weighted sequence