Spline wavelets in \(\ell^{2}(Z)\). (English) Zbl 1093.42023
Summary: Compared with the spline wavelet decomposition for the discrete power growth space \(\mathcal F\) given by A. B. Pevnyi and V. A. Zheludev [J. Approximation Theory 102, No. 2, 286–301 (2000; Zbl 0941.41002); J. Fourier Anal. Appl. 8, No. 1, 59–83 (2002; Zbl 0993.42015)] this paper deals with spline wavelet decompositions for the Hilbert space \(\ell^{2}(Z)\). We characterize RTB splines and RTB wavelets, because the space \(\ell^{2}(Z)\) can be represented by them. It turns out that the representation is stable and the convergence is much stronger than the pointwise convergence in \(\mathcal F\). Finally, a family of symmetric RTB wavelets with finite supports are constructed.
MSC:
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
| 41A15 | Spline approximation |
Keywords:
RTB spline; RTB wavelet; discrete Fourier transform; symmetric wavelet; Riesz basis; discrete splineReferences:
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