Riesz sequences of translates and generalized duals with support on [0, 1]
- 66 Downloads
If the integer translates of a function ø with compact support generate a frame for a subspace W of L 2(ℝ),then it is automatically a Riesz basis for W, and there exists a unique dual Riesz basis belonging to W. Considerable freedom can be obtained by considering oblique duals, i.e., duals not necessarily belonging to W. Extending work by Ben-Artzi and Ron, we characterize the existence of oblique duals generated by a function with support on an interval of length one. If such a generator exists, we show that it can be chosen with desired smoothness. Regardless whether ø is polynomial or not, the same condition implies that a polynomial dual supported on an interval of length one exists.
Math Subject Classifications42C15
Key Words and PhrasesFrames of translates dual frame dual generator short support
Unable to display preview. Download preview PDF.
- de Boor, C.A Practical Guide to Splines, Springer-Verlag, (2001).Google Scholar
- de Boor, C.Splines as Linear Combinations of B-Splines, Approximation Theory, II, (Proc. Internat. Sympos., Univ. Texas, Austin, 1976), 1–47, Academic Press, New York, (1976).Google Scholar
- de Boor, C.On Local Linear Functionals which Vanish at all B-Splines but One, Theory of Approximation, with Applications, (Proc. Conf., Univ. Calgary, Calgary, Alta., 1975), 120–145, Academic Press, New York, (1976).Google Scholar
- Christensen, O.An Introduction to Frames and Riesz Bases, Birkhäuser, (2003).Google Scholar
- Cohen, A., Daubechies, I., and Feauveau, J.-C. Biorthogonal bases of compactly supported wavelets,Comm. Pure Appl. Math., 485–560, (1993).Google Scholar
- Ron, A.Introduction to Shift-Invariant Spaces I: Linear Independence, Multivariate Approximation and Applications, 112–151, Cambridge University Press, Cambridge, (2001).Google Scholar