TY - JOUR
T1 - Orthogonal Exponentials of Planar Self-Affine Measures with Four-Element Digit Set
AU - Li , Hong Guang
JO - Journal of Mathematical Study
VL - 3
SP - 327
EP - 336
PY - 2022
DA - 2022/09
SN - 55
DO - http://doi.org/10.4208/jms.v55n3.22.07
UR - https://global-sci.org/intro/article_detail/jms/20979.html
KW - Infinite orthogonal set, self-affine measure, orthogonal exponentials.
AB - <p style="text-align: justify;">Let $\mu_{{M},{\mathcal{D}}}$ be a self-affine measure generated by an expanding real matrix $M=\left(\begin{array}{cc}a&amp;e\\f&amp;b\end{array} \right)$ and the digit set $\mathcal{D}= \{(0,0)^t, (1,0)^t, (0, 1)^t, (1, 1)^t\}$. In this paper, we consider that when does $L^2(\mu_{M,\mathcal{D}})$ admit an infinite orthogonal set of exponential functions? Moreover, we obtain that if $e=f=0$ and $a, b\in\{\frac{p}{q},p,q\in 2\mathbb{Z}+1\}$, then there exist at most 4 mutually orthogonal exponential functions in $L^2(\mu_{M,\mathcal{D}})$, and the number 4 is the best possible.</p>