Approximation by refinement masks (Q6569606)

Let \(f\) be an arbitrary continuous \(2\pi\)-periodic function satisfying \(f(0)=1\) and \(|f(x)|^2+|f(x+\pi)|^2\le 1\) for all \(x\in \mathbb{R}\). The main result of the paper is Theorem 1 saying that for any \(\varepsilon>0\), there always exists a refinement mask \(m_0\), which is a \(2\pi\)-periodic trigonometric polynomial satisfying \(m_0(0)=1\) and \(|m_0(x)|^2+|m_0(x+\pi)|^2\le 1\), such that \(\|f-m_0\|_{C(\mathbb{R})}<\varepsilon\) and \(m_0\) does not have nontrivial cycles of roots and pairs of symmetric roots. Therefore, there exists a compactly supported Parseval wavelet frame derived from the refinement mask \(m_0\) and its refinable function has stable integer shifts. The proof of the existence of such \(m_0\) is established through several steps of approximation and construction.