${\mathbb R}^d$上的自仿射谱测度和框架谱测度
Self-affine spectral measures and frame spectral measures on ${\mathbb R}^d$
摘要：
研究了整系数${\mathbb R}^d$中仿射迭代函数系生成的不变测度上的Fourier基。我们证明，对于简单的数字集，这些系统满足开集条件，并没有重叠。我们提出了自然的几何条件下，这些措施有一个标准规范正交基或一个框架的指数函数的频率是一个子集的${\mathbb Z}^d$。此外，我们的特点时，这样的措施有一个频谱在${\Bbb Z}^d$。
英文摘要：
We study Fourier bases on invariant measures generated by affine iterated function systems in ${\mathbb R}^d$ with integer coefficients. We show that, for simple digit sets, these systems satisfy the open set condition and have no overlap. We present natural geometric conditions under which such measures have an orthonormal basis or a frame of exponential functions with frequencies being a subset of ${\mathbb Z}^d$. Moreover, we characterize when such measures have a spectrum in ${\Bbb Z}^d$.
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