Applications to analysis: general metatheorems I

In this chapter we show how some of the main results from chapters 8–12 can be combined with the representation of Polish spaces from chapter 4 to establish general metatheorems on the extractability of effective uniform bounds from proofs in analysis. ‘Uniform’ here means the independence of the bounds from parameters in compact metric spaces. In chapter 16 we will apply these results to concrete proofs in best approximation theory and extract effective rates of so-called strong unicity for both best Chebycheff as well as L₁-approximations of continuous functions f∈C[0,1] by polynomials of degree ≤n (and more general so-called Haar spaces in the case of Chebycheff approximation). In chapter 17, the metatheorems will be much generalized to guarantee even strongly uniform bounds which are independent from parameters in abstract bounded metric spaces. These more general theorems will then be applied to proofs in metric fixed point theory in chapter 18.