Analytic elements in p-adic analysis

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. Content: 1. Absolute values and norms -- 2. Infraconnected sets -- 3. Monotonous and circular filters -- 4. Ultrametric absolute values and valuation functions v(h, u) on K(x) -- 5. Hensel Lemma -- 6. Ultrametric field extensions -- 7. Ultraproducts and spherically complete extensions -- 8. A study in [symbol], the p[symbol]-th roots of 1 -- 9. Algebras R(D) -- 10. The analytic elements -- 11. Composition of analytic elements -- 12. Mult(H(D), U[symbol]) -- 13. Power series -- 14. Factorization of analytic elements -- 15. The Mittag-Leffler theorem -- 16. Maximal ideals of codimension 1 -- 17. Dual of a space H(D) -- 18. Algebras H(D) -- 19. Derivative of analytic elements -- 20. Valuation functions for analytic elements -- 21. Elements vanishing along a filter -- 22. Quasi-minorated elements -- 23. Values and zeros of power series -- 24. Quasi-invertible elements -- 25. Zeros theorem for power series -- 26. Image of a disk -- 27. Strictly injective analytic elements -- 28. Logarithm and exponential -- 29. A finite increasing property -- 30. Maximum principle -- 31. Analytic elements meromorphic in a hole -- 32. Motzkin factorization -- 33. Applications of the Motzkin factorization -- 34. Maximum in a circle with holes -- 35. T-filters and T-sequences -- 36. Examples and counter-examples about T-filters -- 37. Characteristic property of the T-filters -- 38. Applications of T-filters -- 39. Integrally closed algebras H(D) -- 40. Absolute values on H(D) -- 41. Distinguished circular filters -- 42. Maximal ideals of infinite codimension -- 43. Idempotent T-sequences -- 44. T-polar sequences -- 45. Analytic extension through a T-filter -- 46. Algebra [symbol] -- 47. Meromorphic products -- 48. Collapsing meromorphic products -- 49. Injectivity, Mittag-Leffler series and Motzkin products -- 50. Analytic functions and analytic elements -- 51. Infinite van der Monde matrices -- 52. p-adic analytic interpolation -- 53. Analytic elements with a zero derivative -- 54. Generalities on the differential equation y' = fy in H(D) -- 55. The differential equation y' = fy in algebras H(D) -- 56. The equation y' = fy in zero residue characteristic -- 57. The equation y' = fy in [symbol] with f not quasi-invertible -- 58. The equation y' = fy in [symbol] with f quasi-invertible -- 59. Residues and equation y' = fy -- 60. Equation g' = fg with [symbol] H(D) -- 61. The p-adic Fourier transform. 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 Discussing the analytic elements in p-adic analysis, this volume covers areas including: ultrametric absolute values and norms; monotonous and circular filters; the ultrametric absolute values on K(x); hensel lemma; and factorization of analytic elements.