Error estimates and reproduction of polynomials for biorthogonal local trigonometric bases (Q1283830)

An approach to biorthogonal local trigonometric bases in the two-overlapping setting was given by \textit{B. Jawerth} and \textit{W. Sweldens} [J. Fourier Anal. Appl. 2, No. 2, 109-133 (1995; Zbl 0886.42024)] and by \textit{C. K. Chui} and \textit{X. Shi} [J. Fourier Anal. Appl. 3, No. 5, 559-575 (1997; Zbl 0913.42026)]. In this paper, error estimates for the approximation with such basis are presented. It turns out that the approximation becomes better for smooth bell and test functions if sufficiently many basis functions exist per interval. Smooth local trigonometric bases are constructed, which reproduce constants resp. linear functions by one resp. a small number of basis functions for each interval. For the two-overlapping setting there does not exist a local trigonometric basis, which can locally reproduce polynomials of degree 2.