Blancmange curve

The periodic version of the Takagi curve can also be defined as the unique bounded solution ${\displaystyle T=T_{w}:\mathbb {R} \to \mathbb {R} }$ to the functional equation

${\displaystyle T(x)=s(x)+wT(2x).}$

Indeed, the blancmange function ${\displaystyle T_{w}}$ is certainly bounded, and solves the functional equation, since

${\displaystyle T_{w}(x):=\sum _{n=0}^{\infty }w^{n}s(2^{n}x)=s(x)+\sum _{n=1}^{\infty }w^{n}s(2^{n}x)}$${\displaystyle =s(x)+w\sum _{n=0}^{\infty }w^{n}s(2^{n+1}x)=s(x)+wT_{w}(2x).}$

Conversely, if ${\displaystyle T:\mathbb {R} \to \mathbb {R} }$ is a bounded solution of the functional equation, iterating the equality one has for any N

${\displaystyle T(x)=\sum _{n=0}^{N}w^{n}s(2^{n}x)+w^{N+1}T(2^{N+1}x)=\sum _{n=0}^{N}w^{n}s(2^{n}x)+o(1),{\text{ for }}N\to \infty ,}$

whence ${\displaystyle T=T_{w}}$. Incidentally, the above functional equations possesses infinitely many continuous, non-bounded solutions, e.g. ${\displaystyle T_{w}(x)+c|x|^{-\log _{2}w}.}$

The blancmange curve can be visually built up out of triangle wave functions if the infinite sum is approximated by finite sums of the first few terms. In the illustrations below, progressively finer triangle functions (shown in red) are added to the curve at each stage.

n = 0	n ≤ 1	n ≤ 2	n ≤ 3

The infinite sum defining ${\displaystyle T_{w}(x)}$ converges absolutely for all ${\displaystyle x}$: since ${\displaystyle 0\leq s(x)\leq 1/2}$ for all ${\displaystyle x\in \mathbb {R} }$, we have:

${\displaystyle \sum _{n=0}^{\infty }|w^{n}s(2^{n}x)|\leq {\frac {1}{2}}\sum _{n=0}^{\infty }|w|^{n}={\frac {1}{2}}\cdot {\frac {1}{1-|w|}}}$ if ${\displaystyle |w|<1.}$

Therefore, the Takagi curve of parameter ${\displaystyle w}$ is defined on the unit interval (or ${\displaystyle \mathbb {R} }$) if ${\displaystyle |w|<1}$.

The Takagi function of parameter ${\displaystyle w}$ is continuous. Indeed, the functions ${\displaystyle T_{w,n}}$ defined by the partial sums ${\displaystyle T_{w,n}(x)=\sum _{k=0}^{n}w^{k}s(2^{k}x)}$ are continuous and converge uniformly toward ${\displaystyle T_{w}}$, since:

${\displaystyle \left|T_{w}(x)-T_{w,n}(x)\right|=\left|\sum _{k=n+1}^{\infty }w^{k}s(2^{k}x)\right|=\left|w^{n+1}\sum _{k=0}^{\infty }w^{k}s(2^{k+n+1}x)\right|\leq {\frac {|w|^{n+1}}{2}}\cdot {\frac {1}{1-|w|}}}$ for all x when ${\displaystyle |w|<1.}$

This value can be made as small as we want by selecting a big enough value of n. Therefore, by the uniform limit theorem, ${\displaystyle T_{w}}$ is continuous if |w| < 1.

• 
  parameter w = 2/3
• 
  parameter w = 1/2
• 
  parameter w = 1/3
• 
  parameter w = 1/4
• 
  parameter w = 1/8

Since the absolute value is a subadditive function so is the function ${\displaystyle s(x)=\min _{n\in {\mathbf {Z} }}|x-n|}$, and its dilations ${\displaystyle s(2^{k}x)}$; since positive linear combinations and point-wise limits of subadditive functions are subadditive, the Takagi function is subadditive for any value of the parameter ${\displaystyle w}$.

For ${\displaystyle w=1/4}$, one obtains the parabola: the construction of the parabola by midpoint subdivision was described by Archimedes.

For values of the parameter ${\displaystyle 0<w<1/2}$ the Takagi function ${\displaystyle T_{w}}$ is differentiable in classical sense at any ${\displaystyle x\in \mathbb {R} }$ which is not a dyadic rational. Precisely, by derivation under the sign of series, for any non dyadic rational ${\displaystyle x\in \mathbb {R} }$ one finds

${\displaystyle T_{w}'(x)=\sum _{n=0}^{\infty }(2w)^{n}\,(-1)^{x_{-n-1}}}$

where ${\displaystyle (x_{n})_{n\in \mathbb {Z} }\in \{0,1\}^{\mathbb {Z} }}$ is the sequence of binary digits in the base 2 expansion of ${\displaystyle x}$, that is, ${\displaystyle x=\sum _{n\in \mathbb {Z} }2^{n}x_{n}}$. Moreover, for these values of ${\displaystyle w}$ the function ${\displaystyle T_{w}}$ is Lipschitz of constant ${\displaystyle 1 \over 1-2w}$. In particular for the special value ${\displaystyle w=1/4}$ one finds, for any non dyadic rational ${\displaystyle x\in [0,1]}$ ${\displaystyle T_{1/4}'(x)=2-4x}$, according with the mentioned ${\displaystyle T_{1/4}(x)=2x(1-x).}$

For ${\displaystyle w=1/2}$ the blancmange function ${\displaystyle T_{w}}$ it is of bounded variation on no non-empty open set; it is not even locally Lipschitz, but it is quasi-Lipschitz, indeed, it admits the function ${\displaystyle \omega (t):=t(|\log _{2}t|+1/2)}$ as a modulus of continuity .

The Takagi–Landsberg function admits an absolutely convergent Fourier series expansion:

${\displaystyle T_{w}(x)=\sum _{m=0}^{\infty }a_{m}\cos(2\pi mx)}$

with ${\displaystyle a_{0}=1/4(1-w)}$ and, for ${\displaystyle m\geq 1}$

${\displaystyle a_{m}:=-{\frac {2}{\pi ^{2}m^{2}}}(4w)^{\nu (m)},}$

where ${\displaystyle 2^{\nu (m)}}$ is the maximum power of ${\displaystyle 2}$ that divides ${\displaystyle m}$. Indeed, the above triangle wave ${\displaystyle s(x)}$ has an absolutely convergent Fourier series expansion

${\displaystyle s(x)={\frac {1}{4}}-{\frac {2}{\pi ^{2}}}\sum _{k=0}^{\infty }{\frac {1}{(2k+1)^{2}}}\cos {\big (}2\pi (2k+1)x{\big )}.}$

By absolute convergence, one can reorder the corresponding double series for ${\displaystyle T_{w}(x)}$:

${\displaystyle T_{w}(x):=\sum _{n=0}^{\infty }w^{n}s(2^{n}x)={\frac {1}{4}}\sum _{n=0}^{\infty }w^{n}-{\frac {2}{\pi ^{2}}}\sum _{n=0}^{\infty }\sum _{k=0}^{\infty }{\frac {w^{n}}{(2k+1)^{2}}}\cos {\big (}2\pi 2^{n}(2k+1)x{\big )}\,:}$

putting ${\displaystyle m=2^{n}(2k+1)}$ yields the above Fourier series for ${\displaystyle T_{w}(x).}$

The recursive definition allows the monoid of self-symmetries of the curve to be given. This monoid is given by two generators, g and r, which act on the curve (restricted to the unit interval) as

${\displaystyle [g\cdot T_{w}](x)=T_{w}\left({\frac {x}{2}}\right)={\frac {x}{2}}+wT_{w}(x)}$

and

${\displaystyle [r\cdot T_{w}](x)=T_{w}(1-x)=T_{w}(x).}$

A general element of the monoid then has the form ${\displaystyle \gamma =g^{a_{1}}rg^{a_{2}}r\cdots rg^{a_{n}}}$ for some integers ${\displaystyle a_{1},a_{2},\cdots ,a_{n}}$ This acts on the curve as a linear function: ${\displaystyle \gamma \cdot T_{w}=a+bx+cT_{w}}$ for some constants a, b and c. Because the action is linear, it can be described in terms of a vector space, with the vector space basis:

${\displaystyle 1\mapsto e_{1}={\begin{bmatrix}1\\0\\0\end{bmatrix}}}$

${\displaystyle x\mapsto e_{2}={\begin{bmatrix}0\\1\\0\end{bmatrix}}}$

${\displaystyle T_{w}\mapsto e_{3}={\begin{bmatrix}0\\0\\1\end{bmatrix}}}$

In this representation, the action of g and r are given by

${\displaystyle g={\begin{bmatrix}1&0&0\\0&{\frac {1}{2}}&{\frac {1}{2}}\\0&0&w\end{bmatrix}}}$

and

${\displaystyle r={\begin{bmatrix}1&1&0\\0&-1&0\\0&0&1\end{bmatrix}}}$

That is, the action of a general element ${\displaystyle \gamma }$ maps the blancmange curve on the unit interval [0,1] to a sub-interval ${\displaystyle [m/2^{p},n/2^{p}]}$ for some integers m, n, p. The mapping is given exactly by ${\displaystyle [\gamma \cdot T_{w}](x)=a+bx+cT_{w}(x)}$ where the values of a, b and c can be obtained directly by multiplying out the above matrices. That is:

${\displaystyle \gamma ={\begin{bmatrix}1&{\frac {m}{2^{p}}}&a\\0&{\frac {n-m}{2^{p}}}&b\\0&0&c\end{bmatrix}}}$

Note that ${\displaystyle p=a_{1}+a_{2}+\cdots +a_{n}}$ is immediate.

The monoid generated by g and r is sometimes called the dyadic monoid; it is a sub-monoid of the modular group. When discussing the modular group, the more common notation for g and r is T and S, but that notation conflicts with the symbols used here.

The above three-dimensional representation is just one of many representations it can have; it shows that the blancmange curve is one possible realization of the action. That is, there are representations for any dimension, not just 3; some of these give the de Rham curves.