Definition
The blancmange function is defined on the unit interval by
![{\displaystyle \operatorname {blanc} (x)=\sum _{n=0}^{\infty }{s(2^{n}x) \over 2^{n)),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a13df8ae007c421961926a9b8cd0bb775ee3da8e)
where
is the triangle wave, defined by
,
that is,
is the distance from x to the nearest integer.
The Takagi–Landsberg curve is a slight generalization, given by
![{\displaystyle T_{w}(x)=\sum _{n=0}^{\infty }w^{n}s(2^{n}x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c58c24f7d4be131ce35e9b68b5675aabf887026c)
for a parameter
; thus the blancmange curve is the case
. The value
is known as the Hurst parameter.
The function can be extended to all of the real line: applying the definition given above shows that the function repeats on each unit interval.
Functional equation definition
The periodic version of the Takagi curve can also be defined as the unique bounded solution
to the functional equation
![{\displaystyle T(x)=s(x)+wT(2x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d197e9e58e52fbcd1e04676c7ef67160d7bef70)
Indeed, the blancmange function
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5dc975bf5d30c9d527062fedba94c0856d35d0c)
![{\displaystyle =s(x)+w\sum _{n=0}^{\infty }w^{n}s(2^{n+1}x)=s(x)+wT_{w}(2x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c596596390d4608addadae43481e7a68fc10098d)
Conversely, if
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 ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da33ca0282cbdc26dc2cbbbe3e78444dbe44f368)
whence
. Incidentally, the above functional equations possesses infinitely many continuous, non-bounded solutions, e.g.
Graphical construction
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.
Properties
Convergence and continuity
The infinite sum defining
converges absolutely for all
Since
for all
![{\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|))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/260020db586c50f7bbc5e95c7ed8727e91ef2a26)
if
The Takagi curve of parameter
is defined on the unit interval (or
) if
. The Takagi function of parameter
is continuous. The functions
defined by the partial sums
![{\displaystyle T_{w,n}(x)=\sum _{k=0}^{n}w^{k}s(2^{k}x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad5c0472ce7a59a269bc812d37682b31285c6fbd)
are continuous and converge uniformly toward
![{\displaystyle {\begin{aligned}\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|))\end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df0c194fef75e19fca95dbefc3c789232ddb95e1)
for all x when
This bound decreases as
By the uniform limit theorem,
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
Subadditivity
Since the absolute value is a subadditive function so is the function
, and its dilations
; since positive linear combinations and point-wise limits of subadditive functions are subadditive, the Takagi function is subadditive for any value of the parameter
.
The special case of the parabola
For
, one obtains the parabola: the construction of the parabola by midpoint subdivision was described by Archimedes.
Differentiability
For values of the parameter
the Takagi function
is differentiable in the classical sense at any
which is not a dyadic rational. By derivation under the sign of series, for any non dyadic rational
one finds
![{\displaystyle T_{w}^{\prime }(x)=\sum _{n=0}^{\infty }(2w)^{n}\,(2b_{n}-1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a546e077dac43e43ac44bf5dd5af72e47593c3f)
where
is the sequence of binary digits in the base 2 expansion of
:
![{\displaystyle x=\sum _{n=-k}^{\infty }b_{n}2^{-n-1}\;.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b139924d7169361bb89cdb92447981c5cb90f20)
Equivalently, the bits in the binary expansion can be understood as a sequence of square waves, the Haar wavelets, scaled to width
This follows, since the derivative of the triangle wave is just the square wave:
![{\displaystyle {\frac {d}{dx))s(x)=\operatorname {sgn}(1/2-(x\!\!\!\mod 1))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae14d8e87cf5bfa8ee92359825367d1484ff0d2d)
and so
![{\displaystyle T_{w}^{\prime }(x)=\sum _{n=0}^{\infty }(2w)^{n}\operatorname {sgn}(1/2-(2^{n}x\!\!\!\mod 1))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68aa2088257dea00816487279c6cf05a3401a986)
For the parameter
the function
is Lipschitz of constant
In particular for the special value
one finds, for any non dyadic rational
, according with the mentioned
For
the blancmange function
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
as a modulus of continuity .
Fourier series expansion
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97fa0470ae9508fb346afe7218e68980e380ce48)
with
and, for
![{\displaystyle a_{m}:=-{\frac {2}{\pi ^{2}m^{2))}(4w)^{\nu (m)},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/933fa72d7e7ee94f6a4d3ef038a27eb30f8754e0)
where
is the maximum power of
that divides
.
Indeed, the above triangle wave
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 )}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86eee7d1e0852e45844e72c3eeb6fee17cbe3526)
By absolute convergence, one can reorder the corresponding double series for
:
![{\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 )}\,:}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da62dc63feb4db6e10f05aff357db094d669f5ac)
putting
yields the above Fourier series for
Self similarity
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(g\cdot x\right)=T_{w}\left({\frac {x}{2))\right)={\frac {x}{2))+wT_{w}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80f034a1c5a31e0450f85eeddfb63b70991e300f)
and
![{\displaystyle [r\cdot T_{w}](x)=T_{w}(r\cdot x)=T_{w}(1-x)=T_{w}(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/288657b954a7fa30be7fc8ce85bfb6e11c5da16b)
A general element of the monoid then has the form
for some integers
This acts on the curve as a linear function:
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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5a418166f8049cf6a1ca1fd65b9273cfc41c3b0)
![{\displaystyle x\mapsto e_{2}={\begin{bmatrix}0\\1\\0\end{bmatrix))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd318921667448e0f503d6f7fb95660ed73af167)
![{\displaystyle T_{w}\mapsto e_{3}={\begin{bmatrix}0\\0\\1\end{bmatrix))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/edb2a585f014d08e536ba00a38aac48a2bc3d7d5)
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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81b3dda4d5df034045724ae8b3bf37f24cf2254f)
and
![{\displaystyle r={\begin{bmatrix}1&1&0\\0&-1&0\\0&0&1\end{bmatrix))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3bbcea371bbd2948b3b1931a0a244802de8b0f7)
That is, the action of a general element
maps the blancmange curve on the unit interval [0,1] to a sub-interval
for some integers m, n, p. The mapping is given exactly by
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))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6e5c1c9b85e8e9a54adef2e09e379f2d78edecd)
Note that
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.
Integrating the Blancmange curve
Given that the integral of
from 0 to 1 is 1/2, the identity
allows the integral over any interval to be computed by the following relation. The computation is recursive with computing time on the order of log of the accuracy required. Defining
![{\displaystyle I(x)=\int _{0}^{x}\operatorname {blanc} (y)\,dy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cad4722d3ae6137ae5d5012eda7daffec5b57ef1)
one has that
![{\displaystyle I(x)={\begin{cases}I(2x)/4+x^{2}/2&{\text{if ))0\leq x\leq 1/2\\1/2-I(1-x)&{\text{if ))1/2\leq x\leq 1\\n/2+I(x-n)&{\text{if ))n\leq x\leq (n+1)\\\end{cases))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2361d08531c4d9ec302e1ce49f04e50973b753f)
The definite integral is given by:
![{\displaystyle \int _{a}^{b}\operatorname {blanc} (y)\,dy=I(b)-I(a).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15e1bf61341a274bcc191d0c618e1470079cf590)
A more general expression can be obtained by defining
![{\displaystyle S(x)=\int _{0}^{x}s(y)dy={\begin{cases}x^{2}/2,&0\leq x\leq {\frac {1}{2))\\-x^{2}/2+x-1/4,&{\frac {1}{2))\leq x\leq 1\\n/4+S(x-n),&(n\leq x\leq n+1)\end{cases))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e12e20d6cbe2d3ce4177a6e094e319dfd1fed28a)
which, combined with the series representation, gives
![{\displaystyle I_{w}(x)=\int _{0}^{x}T_{w}(y)dy=\sum _{n=0}^{\infty }(w/2)^{n}S(2^{n}x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd469d8b6c80c418c77eb4ce332995c45a16ccb)
Note that
![{\displaystyle I_{w}(1)={\frac {1}{4(1-w)))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97c283bb3659b0b0c911639fc0deb500772b3864)
This integral is also self-similar on the unit interval, under an action of the dyadic monoid described in the section Self similarity. Here, the representation is 4-dimensional, having the basis
. The action of g on the unit interval is the commuting diagram
![{\displaystyle [g\cdot I_{w}](x)=I_{w}\left(g\cdot x\right)=I_{w}\left({\frac {x}{2))\right)={\frac {x^{2)){8))+{\frac {w}{2))I_{w}(x).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9b4166fbddbfd1c3d3c2b076e835b0a8d2b5159)
From this, one can then immediately read off the generators of the four-dimensional representation:
![{\displaystyle g={\begin{bmatrix}1&0&0&0\\0&{\frac {1}{2))&0&0\\0&0&{\frac {1}{4))&{\frac {1}{8))\\0&0&0&{\frac {w}{2))\end{bmatrix))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e61417b63e73493572bbce5a8c74b4deac15b76)
and
![{\displaystyle r={\begin{bmatrix}1&1&1&{\frac {1}{4(1-w)))\\0&-1&-2&0\\0&0&1&0\\0&0&0&-1\end{bmatrix))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4feeaa619d8e8765ec67c79fb9095c6988e81d8)
Repeated integrals transform under a 5,6,... dimensional representation.
Relation to simplicial complexes
Let
![{\displaystyle N={\binom {n_{t)){t))+{\binom {n_{t-1)){t-1))+\ldots +{\binom {n_{j)){j)),\quad n_{t}>n_{t-1}>\ldots >n_{j}\geq j\geq 1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d0cb7b8515841839c6abe62b7fbfc4fc53c6759)
Define the Kruskal–Katona function
![{\displaystyle \kappa _{t}(N)={n_{t} \choose t+1}+{n_{t-1} \choose t}+\dots +{n_{j} \choose j+1}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77f3711ee397d0dca4ef94a37815485223572e90)
The Kruskal–Katona theorem states that this is the minimum number of (t − 1)-simplexes that are faces of a set of N t-simplexes.
As t and N approach infinity,
(suitably normalized) approaches the blancmange curve.