Class of smooth nonseparable N-dimensional scaling functions
Móhsen Maesumi · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1994
Wavelets of compact support are an important tool in many areas of signal analysis. A starting point for the construction of such wavelets is the scaling function, the solution of the dilation equation. We study the dilation equation (phi) (X) equals (Sigma) KCK(phi )(2X-K) where K (epsilon) {0...m}N, (phi) :RN yields R, CK (epsilon) R. This paper gives a set of sufficient conditions on CK under which the solution of the dilation equation has a specific degree of regularity. We will construct (phi) through infinite products of 2N associated matrices with entries in terms of CK. The conditions for regularity are based on certain sum rules that triangularize all of the associated matrices and on certain inequalities that control the eigenvalues of the matrices. The net effect of the sum rules is to specify the coefficients CK in terms of a binomial interpolation of their values at the corners of the N-cube, {0,m}N. The inequalities are based on sums of the coefficients at the corners of the various faces of the N-cube. The number of derivatives that (phi) possesses and the Holder exponent of the last derivative can be determined for the sums.