Ten lectures on wavelets
Multidimensional Interpolatory Subdivision Schemes
SIAM Journal on Numerical Analysis
Arbitrarily smooth orthogonal nonseparable wavelets in R2
SIAM Journal on Mathematical Analysis
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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 (1 1 1 -1), 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.