On the holes of a class of bidimensional nonseparable wavelets

Let I be the 2×2 identity matrix, and M a 2×2 dilation matrix with M²=2I. Since one can explicitly construct M-basic wavelets from an MRA related to M, and many applications employ wavelet bases in R², M-wavelets and wavelet frames have been extensively discussed. This paper focuses on dilation matrices M satisfying M²=2I. For any matrix M integrally similar to , an optimal estimate on the boundary of the holes of M-wavelets is obtained. This result tells us the holes cannot be too large. Contrast to this result, when the modulus of the Fourier transform of an M-wavelet is, up to a constant, a characteristic function on some set, a property of this set is obtained, which shows the holes of this kind of wavelets cannot be too small.