 | Jorgensen E.T. Iterated function systems, moments, and tranformations of infinite matrices / P.E.T.Jorgensen, K.A.Kornelson, K.L.Shuman. - Providence: American Mathematical Society, 2011. - ix, 105 p. - (Memoirs of the American Mathematical Society; vol.213, N 1003). - Bibliogr.: p.103-105. - ISBN 978-0-8218-5248-4; ISSN 0065-9266
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Preface ....................................................... vii
Chapter 1. Notation ............................................ 1
1.1 Hilbert space notation ..................................... 1
1.2 Unbounded operators ........................................ 1
1.3 Multi-index notation ....................................... 2
1.4 Moments and moment matrices ................................ 2
1.5 Computations with infinite matrices ........................ 4
1.6 Inverses of infinite matrices .............................. 7
Chapter 2. The moment problem .................................. 9
2.1 The moment problem M = M(μ) ................................ 9
2.2 A Parthasarathy-Kolmogorov approach to the moment
problem ................................................... 12
2.3 Examples .................................................. 16
2.4 Historical notes .......................................... 18
Chapter 3. A transformation of moment matrices: the affine
case ........................................................... 21
3.1 Affine maps ............................................... 21
3.2 IFSs and fixed points of the Hutchinson operator .......... 23
3.3 Preserving Hankel matrix structure ........................ 27
Chapter 4. Moment matrix transformation: measurable maps ...... 31
4.1 Encoding matrix A for τ ................................... 31
4.2 Approximation of A with finite matrices ................... 38
Chapter 5. The Kato-Friedrichs operator ....................... 41
5.1 The quadratic form QM ..................................... 41
5.2 The closability of QM ..................................... 42
5.3 A factorization of the Kato-Friedrichs operator ........... 44
5.4 Kato connection to A matrix ............................... 45
5.5 Examples .................................................. 50
Chapter 6. The integral operator of a moment matrix ........... 53
6.1 The Hilbert matrix ........................................ 53
6.2 Integral operator for a measure supported on [-1, 1] ...... 57
Chapter 7. Boundedness and spectral properties ................ 63
7.1 Bounded Kato operators .................................... 63
7.2 Projection-valued measures ................................ 67
7.3 Spectrum of the Kato operator ............................. 69
7.4 Rank of measures .......................................... 74
7.5 Examples .................................................. 75
Chapter 8. The moment problem revisited ....................... 83
8.1 The shift operator and three incarnations of symmetry ..... 83
8.2 Self-adjoint extensions of a shift operator ............... 85
8.3 Self-adjoint extensions and the moment problem ............ 87
8.4 Jacobi representations of matrices ........................ 90
8.5 The triple recursion relation and extensions to higher
dimensions ................................................ 96
8.6 Concrete Jacobi matrices .................................. 98
Acknowledgements .............................................. 101
Bibliography .................................................. 103
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