Han, Bin; Mo, Qun Tight wavelet frames generated by three symmetric \(B\)-spline functions with high vanishing moments. (English) Zbl 1040.42030 Proc. Am. Math. Soc. 132, No. 1, 77-86 (2004). The motivation for the paper comes from the following question: can one obtain a tight wavelet frame with three generators of compact support arising from any given B-spline \(B_m\) of order \(m\), such that the generators are symmetric and have vanishing moments of (the highest possible) order \(m\)? The main result proves the following. There exist three finitely supported sequences \(b^1,b^2,b^3\) on \(Z\) such that the functions \(\psi^\ell (x)= \sum b^\ell(k) B_m(2x-k), \;\ell=1,2,3\), have the following properties: (i) \(\{\psi^1,\psi^2,\psi^3\}\) generates a tight wavelet frame and has vanishing moments of order \(m\), (ii) \(\psi^1,\psi^2,\psi^3\) are real-valued, symmetric, and compactly supported, and \[ \psi^1(1-t)=(-1)^m\psi^1(t),\quad \psi^2(m-t)= \psi^2(t),\quad \psi^3(m-t)=-\psi^3(t). \] Reviewer: Ole Christensen (Lyngby) Cited in 12 Documents MSC: 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 41A15 Spline approximation Keywords:symmetric tight wavelet frames; \(B\)-spline functions; vanishing moments × Cite Format Result Cite Review PDF Full Text: DOI