Preliminaries and Notation This preliminary chapter fixes notation conventions and normalizations. It also states some basic theorems that will be used later in the book. For those less familiar with Hilbert and Banach spaces, it contains a very brief primer. (This primer should be used mainly as a reference to come back to in those instances when the reader comes across some Hilbert or Banach space language that she or he is unfamiliar with. For most chapters, these concepts are not used. Let us start by some notation conventions. For E E R, we write z for the argest integer not exceeding =max{m∈z;n≤x} For example, 13/2=1,[-3/2=-2, 1-2]=-2 Similarly, [=l is the smallest integer which is larger than or equal to z If a-0(or oo), then we denote by o(a)any quantity that is bounded by a constant times a, by o(a)any quantity that tends to o(or oo) when a does. The end of a proof is always marked with a s; for clarity, many remarks or examples are ended with a o In many proofs, C denotes a "generic"constant, which need not have the same value throughout the proof. In chains of inequalities, I often use C,C,C",……orC1,C2,C3,…… to avoid confusion We use the following convention for the Fourier transform(in one dimension) (万∫)(E)=f()= dxe-1ff(a) (0.0.1) With this normalization, one has )12‖fE 1/p IfM p= dz If(z)IP (0.0.2)