Quantum field theory and a new universal high-energy scale文档.pdfVIP

Chapter 3 Weierstrass-Type Functions I Summary. The aim of this chapter is to present various classical methods of testing the nowhere differen- tiability of the Weierstrass-type function x −→ ∞ an cosp (2πbnx + θn ). More developed results will be n=0 discussed in Chap. 8. 3.1 Introduction We will discuss the nowhere differentiability of the following Weierstrass-type function ∞ W (x) := an cosp (2πbnx + θ ), x ∈ R, (3.1.1) p,a,b,θ n n=0 where p ∈ N, 0 a 1, ab ≥ 1, θ := (θn )∞ ⊂ R. (3.1.2) n=0 Throughout the chapter, we always assume that p, a, b, θ satisfy (3.1.2) (cf. Figs. 3.1, 3.2, and 3.3). Notice that the function W with p = 1, b ∈ 2N + 1, and ab 1 + 3 π , coincides with 1,a,b,0 2 the original nowhere differentiable Weierstrass function presented by him to the K¨onigliche Akademie der Wissenschaften on 18 July 1872; cf. [Wei86]. We will be mainly interested in a characterization of the parameters p, a, b, θ for which the function W belongs to one of the following three classes of nowhere differentiable p,a,b,θ functions: ND∞ (R), ND± (R), and M(R) ∩ ND∞ (R). Recall that M(R) ⊂ ND± (R). We would like to point out that in general, most of the cases are not completely understood (even for p = 1 and θ = 0). To simplify notation, we will use the following conventions: • If θn = θ for all n ∈ N0 , then we simply