Spaces of Analytic Functions

We present here spaces of analytic functions A α , d n ⊂ S n $$\mathbf{A}_{\alpha,d}^{n} \subset \mathbf{S}^{n}$$ as well as spaces, A α , d , T n ⊂ S T n $$\mathbf{A}_{\alpha,d,T}^{n} \subset \mathbf{S}_{T}^{n}$$ , n = 1, 2, 3. In this chapter, we shall study the properties of these spaces, we shall prove in Chap. 3 that if the components of the initial condition vector u 0 belong to A α, d 3 then each component of N u of ( 1.23 ) belongs to A α, d, T 3, and we shall furthermore prove in Chap. 4 that the solution to ( 1.23 ) belongs to A α, d, T 3, for all T sufficiently small. These spaces are in fact special cases of the spaces S n and S T n introduced in Sect. 1.2 They provide several conveniences, such as enabling sharper error bounds and yielding exponential convergence of our approximate solution which we obtain in Chap. 5