Examples of refinable componentwise polynomials
A function \(\phi: {\mathbb R} \to {\mathbb C}\) is called a componentwise polynomal if there exists an open set \(G\) such that the Lebesgue measure of \({\mathbb R}\setminus G\) is zero and the restriction of \(\varphi\) to every connected component of \(G\) is a polynomial. A function \(\varphi\) is called interpolatory if it is continuous, \(\varphi (0)=1\), and \(\varphi (k)=0\) for all integer \(k \neq 0\). A function \(\varphi\) is called orthonormal if \(\{\varphi(\cdot-k)\}\) is an orthonormal system in \(L_2({\mathbb R})\). A function \(\varphi\) is called \(M\)-refinable if \({\widehat \varphi}(M\xi)=H(\xi) {\widehat \varphi}(\xi)\), where \({\widehat \varphi}\) is a Fourier transform of \(\varphi\) and \(H\) is a \(2\pi\)-periodic trigonometric polynomial. The paper presents four examples of compactly supported orthonormal refinable componentwise polynomial functions. In these examples \(M\) equals 3, 6, or 9. The authors note that for \(M=2\) a compactly supported refinable componentwise polynomial must be a B-spline.
scientific article; zbMATH DE number 5148788