Journal of the
Korean Mathematical Society JKMS

In this paper, we establish a number system in $\mathbb{R}^{n}$ which arises from a Haar wavelet basis in connection with decompositions of certain Cuntz algebra representations on $L^2 ( \mathbb R^n) $. Number systems in $\mathbb{R}^{n}$ are also of independent interest [9]. We study radix-representations of $x\in\mathbb{R}^{n}$: \[ x:=a_{l}a_{l-1}\cdots a_{1}a_{0}\cdot a_{-1}a_{-2}\cdots \] as \[x=M^{l}a_{l}+\cdots+Ma_{1}+a_{0}+M^{-1}% a_{-1}+M^{-2}a_{-2}+\cdots \] where each $a_{k}\in D$, and $D$ is some specified digit set. Our analysis uses iteration techniques of a number-theoretic flavor. The viewpoint is a dual one which we term "fractals in the large vs. fractals in the small," illustrating the number theory of integral lattice points vs. "fractions".

Keywords: $C^{\ast}$-algebra, radix-representation, representation of $c^{*}$-algebra, wavelet basis, fractal

MSC numbers: Primary 11A63; Secondary 46L45