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.