Cube-approximating bounded wavelet sets in $\mathbb {R}^{n}$

Abstract:

We prove that for any real expansive $n\times n$ matrix $A$, there exists a bounded $A$-dilation wavelet set in the frequency domain $\mathbb {R}^{n}$ (the inverse Fourier transform of whose characteristic function is a band-limited single wavelet in the time domain $\mathbb {R}^{n}$). Moreover these wavelet sets can approximate a cube in $\mathbb {R}^{n}$ arbitrarily. This result improves Dai, Larson and Speegle’s result about the existence of (basically unbounded) wavelet sets for real expansive matrices.