Takagi–Landsberg curve: Difference between revisions
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:<math> T_w(x) = \sum_{n=0}^\infty w^n s(2^{n}x)</math> | :<math> T_w(x) = \sum_{n=0}^\infty w^n s(2^{n}x)</math> | ||
for a parameter w; thus the blancmange curve is the case <math>w = 1 / 2</math>. The value <math>H = − log 2w</math> is known as the Hurst parameter. For <math>w = 1 / 4</math>, one obtains the parabola: the construction of the parabola by midpoint subdivision was described by Archimedes. | for a parameter w; thus the blancmange curve is the case <math>w = 1 / 2</math>. The value <math>H=−\log 2w</math> is known as the Hurst parameter. For <math>w = 1 / 4</math>, one obtains the parabola: the construction of the parabola by midpoint subdivision was described by Archimedes. | ||
Revision as of 17:26, 18 March 2009
The blancmange function is defined on the unit interval by
- [math]\displaystyle{ {\rm blanc}(x) = \sum_{n=0}^\infty {s(2^{n}x)\over 2^n}, }[/math]
where [math]\displaystyle{ s(x) }[/math] is defined by [math]\displaystyle{ s(x)=\min_{n\in{\bold Z}}|x-n| }[/math], that is, [math]\displaystyle{ s(x) }[/math] is the distance from x to the nearest integer. The infinite sum defining [math]\displaystyle{ blanc(x) }[/math] converges absolutely for all x, but the resulting curve is a fractal. The blancmange function is continuous but nowhere differentiable.
The Takagi–Landsberg curve is a slight generalization, given by
- [math]\displaystyle{ T_w(x) = \sum_{n=0}^\infty w^n s(2^{n}x) }[/math]
for a parameter w; thus the blancmange curve is the case [math]\displaystyle{ w = 1 / 2 }[/math]. The value [math]\displaystyle{ H=−\log 2w }[/math] is known as the Hurst parameter. For [math]\displaystyle{ w = 1 / 4 }[/math], one obtains the parabola: the construction of the parabola by midpoint subdivision was described by Archimedes.
