Two recent results on wavelets: Wavelet bases for the interval, and biorthogonal wavelets diagonalizing the derivative operator. (English) Zbl 0829.42021

Schumaker, Larry L. (ed.) et al., Recent advances in wavelet analysis. Boston, MA: Academic Press, Inc. Wavelet Anal. Appl. 3, 237-257 (1994).
The authoress discusses recent results concerning two problems. First, she talks about how to adapt a wavelet basis on \(\mathbb{R}\) to a wavelet basis on the interval \([0, 1]\) while avoiding problems at the boundary. She reviews the construction of Y. Meyer [Rev. Mat. Iberoam. 7, No. 2, 115-133 (1991; Zbl 0753.42015)] and presents a number of other constructions, including a new one proposed independently by Jawerth, by Jouini and Lemari√©, and by A. Cohen, I. Daubechies and P. Vial [Appl. Comput. Harmon. Anal. 1, No. 1, 54-81 (1993; Zbl 0795.42018)]. This new construction starts from the \(N\) vanishing moment family of I. Daubechies [Commun. Pure Appl. Math. 41, No. 7, 901- 996 (1988; Zbl 0644.42026)], translates the functions and retains the interior scaling functions while adding adapted edge scaling functions so that their union still generates the polynomials on \([0, 1]\) up to a certain degree. The second question is concerned with the problem that the derivative operator is not diagonal in a wavelet basis. Again she outlines several approaches to this problem, emphasizing the solution given by A. Cohen, I. Daubechies and J.-C. Feauveau [Commun. Pure Appl. Math. 45, No. 5, 485-560 (1992; Zbl 0776.42020)]. – For some reason the second page of the bibliography was not printed, so that precise references to items [11] through [18] cannot be found.
For the entire collection see [Zbl 0782.00090].
Reviewer: J.S.Joel (Kelly)


42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces