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Concept
Blancmange curve
Summary
In mathematics, the blancmange curve is a self-affine curve constructible by midpoint subdivision. It is also known as the Takagi curve, after Teiji Takagi who described it in 1901, or as the Takagi–Landsberg curve, a generalization of the curve named after Takagi and Georg Landsberg. The name blancmange comes from its resemblance to a Blancmange pudding. It is a special case of the more general de Rham curve; see also fractal curve.
The blancmange function is defined on the unit interval by
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
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.
The function could also be defined by the series in the section Fourier series expansion.
The periodic version of the Takagi curve can also be defined as the unique bounded solution to the functional equation
Indeed, the blancmange function is certainly bounded, and solves the functional equation, since
Conversely, if is a bounded solution of the functional equation, iterating the equality one has for any N
whence . Incidentally, the above functional equations possesses infinitely many continuous, non-bounded solutions, e.g.
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.
The infinite sum defining converges absolutely for all : since for all , we have:
if
Therefore, the Takagi curve of parameter is defined on the unit interval (or ) if .
The Takagi function of parameter is continuous. Indeed, the functions defined by the partial sums are continuous and converge uniformly toward , since:
for all x when
This value can be made as small as we want by selecting a big enough value of n.
Official source
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