Abstract:
Construction of compactly supported tight wavelet frames in the multivariate case is much more complicated than in the univariate case because an analog of the Riesz Lemma does not exist for the trigonometric polynomials of several variables. We give a new method for the construction of compactly supported tight wavelet frames in $L_2({\Bbb R}^d)$ with any preassigned approximation order $n$ for arbitrary matrix dilation $M$. The number of wavelet functions generating a frame constructed in this way is less or equal to $(d+1)|\det M|-d$. Earlier Bin Han proposed another method, where the number of generating wavelet functions is less than $\left(\frac32\right)^d |\det M|$. Another advantage of our method is in its simplicity. The method is algorithmic, and the algorithm is simple to use, it consists mainly of explicit formulas. Computations are needed only to find several trigonometric polynomials of one variable from their squared magnitudes. The number of generating wavelet functions can be reduced for a large enough class of matrices $M$. Namely, if all entries of some column of $M$ are divisible by $|\det M|$, then the algorithm can be simplified so that the number of wavelet functions does not exceed $|\det M|$. Moreover, the existence of compactly supported tight wavelet frames with $|\det M|-1$ wavelet functions and an arbitrary approximation order is proved for such matrices.