Fractal Functions and Schauder Bases

1. Introduction. In recent years more and more attention has been paid in mathematical papers to fractal functions and to fractal sets. There are various definitions of those objects. We assume that a compact set K ∈ R d+1 is fractal, by definition, if its box (entropy) dimension dim b (K) = j for j = 0, 1,. .. , d + 1 and 0 < dim b (K) < d + 1. At the same time the function f : I d → R d , I = [0, 1], is fractal, by definition, if its graph Γ f = {(t, f (t)) : t ∈ I d } has box dimension satisfying the inequalities d < dim b (Γ f) < d+1. For the definitions and properties of lower dim b (K) and upper dim b (K) box (counting) dimension we refer to [F]. In the case dim b (K)=dim b (K), dim b (K) is by definition the common value. The relation between box dimension of the graph of a function and its Hölder exponent is known for years. In particular, it is known that the Hölder condition with some α, 0 < α ≤ 1, i.e.