Editing Dyadic transformation (section)

===Spectrum===
To obtain the spectrum of <math>\mathcal{L}_T</math>, one must provide a suitable set of [[basis function]]s for the space <math>\mathcal{F}.</math> One such choice is to restrict <math>\mathcal{F}</math> to the set of all polynomials. In this case, the operator has a [[discrete spectrum]], and the [[eigenfunction]]s are (curiously) the [[Bernoulli polynomial]]s!<ref>Pierre Gaspard, "''r''-adic one-dimensional maps and the Euler summation formula", ''Journal of Physics A'', '''25''' (letter) L483-L485 (1992).</ref> (This coincidence of naming was presumably not known to Bernoulli.)

Indeed, one can easily verify that
:<math>\mathcal{L}_T B_n= 2^{-n}B_n</math>
where the <math>B_n</math> are the [[Bernoulli polynomials]]. This follows because the Bernoulli polynomials obey the identity
:<math>\frac{1}{2}B_n\!\left(\frac{y}{2}\right) + \frac{1}{2}B_n\!\left(\frac{y+1}{2}\right) = 2^{-n}B_n(y)</math>
Note that <math>B_0(x)=1.</math>

Another basis is provided by the [[Haar basis]], and the functions spanning the space are the [[Haar wavelet]]s. In this case, one finds a [[continuous spectrum]], consisting of the unit disk on the [[complex plane]]. Given <math>z\in\mathbb{C}</math> in the unit disk, so that <math>|z|<1</math>, the functions

:<math>\psi_{z,k}(x)=\sum_{n=1}^\infty z^n \exp i\pi(2k+1)2^nx</math>
obey
:<math>\mathcal{L}_T \psi_{z,k}= z\psi_{z,k}</math>
for <math>k\in\mathbb{Z}.</math> This is a complete basis, in that every [[integer]] can be written in the form <math>(2k+1)2^n.</math> The Bernoulli polynomials are recovered by setting <math>k=0</math> and <math>z=\frac{1}{2}, \frac{1}{4}, \dots</math>

A complete basis can be given in other ways, as well; they may be written in terms of the [[Hurwitz zeta function]]. Another complete basis is provided by the [[Takagi function]]. This is a fractal, [[nowhere differentiable|differentiable-nowhere]] function. The eigenfunctions are explicitly of the form

:<math>\mbox{blanc}_{w,k}(x) = \sum_{n=0}^\infty w^n s((2k+1)2^{n}x)</math>
where <math>s(x)</math> is the [[triangle wave]]. One has, again,
:<math>\mathcal{L}_T \mbox{blanc}_{w,k} = w\;\mbox{blanc}_{w,k}.</math>

All of these different bases can be expressed as linear combinations of one-another. In this sense, they are equivalent. 

The fractal eigenfunctions show an explicit symmetry under the fractal [[groupoid]] of the [[modular group]]; this is developed in greater detail in the article on the [[Takagi function]] (the blancmange curve). Perhaps not a surprise; the Cantor set has exactly the same set of symmetries (as do the [[continued fraction]]s.) This then leads elegantly into the theory of [[Elliptic partial differential equation|elliptic equation]]s and [[modular form]]s.