A Continuous Analog of the 1D Thue-Morse Sequence
A Continuous Analog of the 1D Thue-Morse Sequence
This Demonstration shows a connection [1, 2] between the Thue–Morse (Prouhet–Thue–Morse) sequence and the infinitely differentiable or atomic functions. By definition, an atomic function is a finite solution of a functional differential equation (FDE) of advanced type, such as , where and is a linear differential operator with constant coefficients. The simplest example of such an FDE is , with . Its solution is called the (or ) function with support .
{1,-1}
up(x)
hut(x)
ℒf(x)=λf(ax+)
n
Σ
k=1
c
k
b
k
|a|>1
ℒ
f'(x)=2f(2x+1)-2f(2x-1)
f(0)=1
∞
C
up(x)
hut(x)
[-1,1]
This function was found independently by Vladimir L. Rvachev, Vladimir A. Rvachev, and Wolfgang Hilberg in 1971. That was the beginning of research into the theory of atomic functions theory. Later, Wolfgang Hilberg found an extremely bizarre connection between the function and the Thue–Morse sequence [4–6], which allows the calculation of the function (red curve in the plot) by iterations of the substitution system that generates the Thue–Morse sequence (filled rectangular pulses in the plot). Such a connection helps in the study of both atomic function theory (a branch of functions) and Thue–Morse-like sequences. It takes advantage of numerical methods, digital signal processing, and many other applications of atomic functions and Thue–Morse sequences [2].
up(x)
up(x)
∞
C