Best Approximation by Analytic and Meromorphic Functions

Abstract

Let φ be a function on 𝕋 of class BMO. As we have already discussed in Chapter 1, there exists a function f ∈ BMOA such that φ — f ∈L^∞(𝕋) and

$$\left\| {\varphi - f} \right\|{L^{\infty }} = \left\| {{H_{\varphi }}} \right\| $$

((0.1))

Such a function f is called a best approximation to φ by analytic functions in L^∞. We have already seen in §1.1 that in general a best approximation is not unique (see Theorem 5.1.5, which gives a necessary and sufficient condition for uniqueness in terms of the Hankel operator Hφ).