We consider the sum of digits functions for both base phi, and for the Zeckendorf expansion of the natural numbers. For both sum of digits functions we present morphisms on infinite alphabets such that these functions viewed as infinite words are letter-to-letter projections of fixed points of these morphisms. We characterize the first differences of both functions a) with generalized Beatty sequences, or unions of generalized Beatty sequences, and b) with morphic sequences.

Morphic words are letter-to-letter images of fixed points x of morphisms on finite alphabets. There are situations where these letter-to-letter maps do not occur naturally, but have to be replaced by a morphism. We call this a decoration of x. Theoretically, decorations of morphic words are again morphic words, but in several problems the idea of decorating the fixed point of a morphism is useful. We present two of such problems. The first considers the so called AA sequences, where α is a quadratic irrational, A is the Beatty sequence defined by A(n)=⌊αn⌋, and AA is the sequence (A(A(n))). The second example considers homomorphic embeddings of the Fibonacci language into the integers, which turns out to lead to generalized Beatty sequences with terms of the form V(n)=p⌊αn⌋+qn+r, where p,q and r are integers.

Polygonal two-dimensional curves are coded by means of a finite number of their Fourier coefficients and then reconstructed by the corresponding Fourier sums. This mechanism is studied for an interesting class of contours whose members exhibit detail at a wide range of scales. Two bounds (one cheaply computed, one with high performance) are given on the number of Fourier coefficients needed to obtain a given accuracy in a reconstruction.