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Introduction | 1 | |
I | Linear approximation in convex metric spaces | 5 |
1 | Convex metric spaces | 5 |
1.1 | Convex spaces | 6 |
1.2 | Convex metric spaces | 8 |
2 | Decomposability in totally convex metric spaces | 14 |
2.1 | Integration in totally convex metric spaces | 14 |
2.2 | Decomposability of elements of a totally convex metric space | 16 |
3 | Special cases | 20 |
3.1 | The set of weight functions is the set of discrete probability distribution functions with jumps at finitely many prescribed points | 20 |
3.2 | The set of weight functions is the set of discrete probability distribution functions with jumps at infinitely many prescribed points | 23 |
3.3 | The set of weight functions is the set of absolutely continuous probability distribution functions with square integrable density function | 29 |
II | Decomposability of distribution functions | 35 |
4 | Formulation of the problem | 35 |
5 | Decomposability of distribution functions | 49 |
5.1 | Considerations on the set of all probability distribution functions | 50 |
5.2 | Considerations on the set of probability distribution functions concentrated on a finite or infinite interval | 63 |
5.3 | Considerations on the set of continuous probability distribution functions | 70 |
5.4 | Considerations on the set of discrete probability distribution functions | 84 |
App. A Two theorems on mixtures of probability distribution functions | 97 | |
App. B Totally positive matrices. The transsignation of a matrix | 100 | |
App. C The determinant theorem of Cauchy, with corollaries | 101 | |
App. D A representation of two polynomials | 105 | |
App. E Cauchy matrices of special type | 107 | |
App. F On a matrix identity | 110 | |
App. G Solution of a matrix equation. On an extension of the Sherman-Morrison theorem | 112 | |
App. H The moment problem of Hamburger. On the solution of the full moment problem of Stieltjes | 116 | |
App. J Integration by parts for Stieltjes integrals | 123 | |
References and bibliography | 129 | |
Index | 132 |
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