Irregular scaling functions with orthogonal translations ✩

For all integers m,k > 1 with m ≡ 1modk we construct, among others, a function f :R→R with dense graph in the set [0, k] × R such that f (x) = m−1 n=0 f (mx −kn) for every x ∈ R, i∈Z f (x + i) = 1 for almost every x ∈ R, and f (x)f (x − i) = 0 for almost every x ∈ R and every i ∈ Z \ {0}.  2005 Elsevier Inc. All rights reserved