Chapter 7: A Partition of Unity and Taylor's Formula

It is convenient to devote a chapter to several technical results which will be of particular importance for us in the next two chapters. In Theorem 7.1 we construct a partition of unity. Then we use this partition of unity to decompose a symbol σ(x, ξ) into a family {σ_k(x, ξ)} of symbols with compact support in the ξ variable. We are able to obtain good estimates on the partial Fourier transforms (with respect to the ξ variable) of all the symbols σ_k(x, ξ). The precise estimates are given in Theorem 7.2. In Theorem 7.3 we prove a multi-dimensional version of Taylor's formula with integral remainder…