New stable biorthogonal spline-wavelets on the interval
In this interesting paper, the author constructs new stable biorthogonal spline-wavelet bases on \([0,\,1]\). This construction is based on the biorthogonal multiresolution analysis (MRA) on \(\mathbb R\) presented by \textit{A. Cohen}, \textit{I. Daubechies}, and \textit{J.-C. Feauveau} [Commun. Pure Appl. Math. 45, No. 5, 485--560 (1992; Zbl 0776.42020)]. For a first construction of biorthogonal spline-wavelets on \([0,\,1]\) [see \textit{W. Dahmen}, \textit{A. Kunoth}, and \textit{K. Urban}, Appl. Comput. Harm. Anal. 6, No. 2, 132--196 (1999; Zbl 0922.42021)]. The aim of this new approach to biorthogonal spline-wavelets on \([0,\,1]\) are better Riesz bounds and better condition numbers of the corresponding wavelet transform matrices. For the construction of the primal MRA, the author chooses the Schoenberg spline bases on \([0,\,1]\) concerning equidistant, dyadic partitions of \([0,\,1]\). The full degree of polynomial reproduction is preserved on the primal side. The construction of boundary scaling functions on the dual side guarantees polynomial reproduction on the dual side too. The MRA spaces satisfy certain Jackson and Bernstein inequalities such that the associated wavelets are \(L^2([0,\,1])\)-stable with nice Riesz bounds. The biorthogonal wavelets are computed by the method of stable completion. The corresponding decomposition and reconstruction transforms are sparse matrices with low condition numbers. The author derives also biorthogonal spline-wavelet bases on \([0,\,1]\) with complementary or homogeneous boundary conditions.
scientific article; zbMATH DE number 5703374
scientific article