Some sufficient conditions for the convergence of the cascade algorithm and for the continuity of the scaling function

Abstract

We define a class of matrices which includes, under some natural assumptions, the matrices \(\mathbf{m}\left( 0\right)\), \(\mathbf{m}\left( 1\right)\) and \(T_{2N-1}\), which are the key matrices of the wavelets theory. The matrices of this class have the property that the eigenvalues of a product matrix are products of their eigenvalues. This property is used in establishing some sufficient conditions for the convergence of the cascade algorithm and some sufficient conditions for the continuity of the scaling function. We generalize here the particular results obtained by us in a previous paper.

Keywords

• dilation equation
• scaling function
• cascade algorithm
• wavelets