Orthogonal polynomial wavelets
The authors construct orthogonal polynomial wavelets and extend some results of \textit{B. Fischer} and \textit{J. Prestin} [Math. Comput. 66, No. 220, 1593-1618 (1997; Zbl 0896.42020)]. Let \(P_j\) \((j= 0,1,\dots)\) be orthonormal polynomials on \([a,b]\) \((-\infty\leq a< b\leq\infty)\) with respect to a nonnegative weight function \(w\). Using the kernel polynomial \[ K_n(t,\xi)= \sum^n_{j=0} P_j(t) P_j(\xi), \] scaling functions and wavelets are introduced by \[ \varphi_{n,r}(t)= K_n(t, x^{(n+1)}_r)\qquad (r= 0,\dots, n), \] \[ \psi_{n,r}(t)= K_{2n}(t, z^{(n)}_r)- K_n(t, z^{(n)}_r)\qquad (r= 0,\dots, n-1), \] where \(x^{(n+1)}_0< x^{(n+1)}_1<\cdots< x^{(n+1)}_n\) and \(z^{(n)}_0< z^{(n)}_1<\cdots z^{(n)}_{n-1}\) are convenient parameters. The authors characterize those parameters \(z^{(n)}_r\) which lead to orthogonal wavelets. The corresponding decomposition and reconstruction schemes are sketched. Finally, the special case of Bernstein-Szegő weight function \(w\) is discussed.
scientific article; zbMATH DE number 1764839