Explicitly given pairs of dual frames with compactly supported generators and applications to irregular B-splines.(English)Zbl 1208.41003

Summary: We consider systems of functions appearing by letting a class of modulations act on a countable collection of functions. These systems correspond to shift-invariant systems, considered on the Fourier side. We provide sufficient conditions for the system to be a frame, as well as an explicit construction of a class of frames and associated duals. We use the result to construct frames based on B-splines with knot sequences satisfying a natural condition, as well as explicitly given duals.

MSC:

 41A15 Spline approximation 42C15 General harmonic expansions, frames
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References:

 [1] de Boor, C., A practical guide to splines, (2001), Springer New York · Zbl 0987.65015 [2] Christensen, O., An introduction to frames and Riesz bases, (2003), Birkhäuser Boston · Zbl 1017.42022 [3] Christensen, O., Pairs of dual Gabor frame generators with compact support and desired frequency localization, Appl. comput. harmon. anal., 20, 403-410, (2006) · Zbl 1106.42030 [4] Goodman, T.N.; Lee, S.L., Wavelets of multiplicity r, Trans. amer. math. soc., 342, 307-324, (1994) · Zbl 0799.41013 [5] Janssen, A.J.E.M., The duality condition for Weyl-Heisenberg frames, () · Zbl 0890.42006 [6] Massopust, P.R.; Ruch, D.K.; Van Fleet, P.J., On the support properties of scaling vectors, Appl. comput. harmon. anal., 3, 229-238, (1996) · Zbl 0858.42023 [7] Ron, A.; Shen, Z., Frames and stable bases for shift-invariant subspaces of $$L^2(\mathbb{R}^d)$$, Canad. J. math., 47, 5, 1051-1094, (1995) · Zbl 0838.42016 [8] Ron, A.; Shen, Z., Weyl – heisenberg systems and Riesz bases in $$L^2(\mathbb{R})$$, Duke math. J., 89, 237-282, (1997) · Zbl 0892.42017 [9] Ron, A.; Shen, Z., Affine systems in $$L_2(R^d)$$: dual systems, J. Fourier anal. appl., 3, 617-637, (1997) · Zbl 0904.42025 [10] Schumaker, L., Spline functions: basic theory, (1981), Wiley-Interscience Boston · Zbl 0449.41004 [11] Walnut, D., Continuity properties of the Gabor frame operator, J. math. anal. appl., 165, 479-504, (1992) · Zbl 0763.47014
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