Book contents
- Frontmatter
- Contents
- Prelude
- Dependence chart
- 1 Prologue
- 2 The pleasures of counting
- 3 σ-algebras
- 4 Measures
- 5 Uniqueness of measures
- 6 Existence of measures
- 7 Measurable mappings
- 8 Measurable functions
- 9 Integration of positive functions
- 10 Integrals of measurable functions and null sets
- 11 Convergence theorems and their applications
- 12 The function spaces Lp, 1 ≤ p ≤ ∞
- 13 Product measures and Fubini's theorem
- 14 Integrals with respect to image measures
- 15 Integrals of images and Jacobi's transformation rule
- 16 Uniform integrability and Vitali's convergence theorem
- 17 Martingales
- 18 Martingale convergence theorems
- 19 The Radon–Nikodým theorem and other applications of martingales
- 20 Inner product spaces
- 21 Hilbert space h
- 22 Conditional expectations in L2
- 23 Conditional expectations in Lp
- 24 Orthonormal systems and their convergence behaviour
- Appendix A lim inf and lim sup
- Appendix B Some facts from point-set topology
- Appendix C The volume of a parallelepiped
- Appendix D Non-measurable sets
- Appendix E A summary of the Riemann integral
- Further reading
- References
- Notation index
- Name and subject index
24 - Orthonormal systems and their convergence behaviour
Published online by Cambridge University Press: 05 September 2012
- Frontmatter
- Contents
- Prelude
- Dependence chart
- 1 Prologue
- 2 The pleasures of counting
- 3 σ-algebras
- 4 Measures
- 5 Uniqueness of measures
- 6 Existence of measures
- 7 Measurable mappings
- 8 Measurable functions
- 9 Integration of positive functions
- 10 Integrals of measurable functions and null sets
- 11 Convergence theorems and their applications
- 12 The function spaces Lp, 1 ≤ p ≤ ∞
- 13 Product measures and Fubini's theorem
- 14 Integrals with respect to image measures
- 15 Integrals of images and Jacobi's transformation rule
- 16 Uniform integrability and Vitali's convergence theorem
- 17 Martingales
- 18 Martingale convergence theorems
- 19 The Radon–Nikodým theorem and other applications of martingales
- 20 Inner product spaces
- 21 Hilbert space h
- 22 Conditional expectations in L2
- 23 Conditional expectations in Lp
- 24 Orthonormal systems and their convergence behaviour
- Appendix A lim inf and lim sup
- Appendix B Some facts from point-set topology
- Appendix C The volume of a parallelepiped
- Appendix D Non-measurable sets
- Appendix E A summary of the Riemann integral
- Further reading
- References
- Notation index
- Name and subject index
Summary
In Chapter 21 we discussed the importance of orthonormal systems (ONSs) in Hilbert spaces. In particular, countable complete ONSs turned out to be bases of separable Hilbert spaces. We have also seen that a countable ONS gives rise to a family of finite-dimensional subspaces and a sequence of orthogonal projections onto these spaces. In the present chapter we are concerned with the following topics:
to give concrete examples of (complete) ONSs;
to see when the associated canonical projections are conditional expectations;
to understand the Lp (p ≠ 2) and a.e. convergence behaviour of series expansions with respect to certain ONSs.
The latter is, in general, not a trivial matter. Here we will see how we can use the powerful martingale machinery of Chapters 17 and 18 to get Lp (1 ≤ p < ∞) and a.e. convergence.
Throughout this chapter we will consider the Hilbert space L2(I, B(I), ρλ) where I ⊂ ℝ is a finite or infinite interval of the real line, B(I) = I ∩ B(ℝ) are the Borel sets in I, λ = λ1|I is Lebesgue measure on I and ρ(x) is a density function. We will usually write ρ(x) dx and ∫ … dx instead of ρλ and ∫ … dλ.
- Type
- Chapter
- Information
- Measures, Integrals and Martingales , pp. 276 - 312Publisher: Cambridge University PressPrint publication year: 2005