Refinement differential equations and wavelets (Q1289312)

The author studies the refinement differential equation (RDE) of type \((P,H)\), \[ P(D)\phi (x)=2[H(E)\phi ](2x), \] where \(P(\lambda)\) is a polynomial in \(\lambda\), \(H(z)\) is a Laurent polynomial in \(z\), both with real coefficients, \(D=d /dx\) and \(E\) denotes the translation operator \(Ef(x)=f(x+1)\). The interesting results are derived for RDEs of type \((P(\lambda),1)\). New functions \(\text{kam}(x)\) and \(\Phi_{\theta}(x)\) are introduced as the atomic solutions to RDEs of type \((P(\lambda),1)\) when \(P(\lambda)\) has no purely imaginary roots. When \(P(\lambda)\) contains at least one purely imaginary root, it is possible to establish a set of linearly independent, periodic and \(C^{\infty}\) solutions based on the well-adapted structure of Dirichlet series. The general regular RDEs of type \((P(\lambda),H(z))\) are discussed as well. The main result is that a scaling function designed by the \((1,H(z))\) refinement equation is smoothed by the \((P(\lambda),1)\) RDE to yield a \(C^{\infty}\) solution to the original \((P(\lambda),H(z))\) RDE. The author discusses some related topics, including the probabilistic method, the generalized subdivision process and the application in the wavelet theory.