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  • Proceedings of the Edinburgh Mathematical Society, Volume 36, Issue 1
  • February 1993, pp. 69-85

On linear independence for integer translates of a finite number of functions

  • Rong-Qing Jia (a1) and Charles A. Micchelli (a2)
  • DOI:
  • Published online: 01 January 2009

We investigate linear independence of integer translates of a finite number of compactly supported functions in two cases. In the first case there are no restrictions on the coefficients that may occur in dependence relations. In the second case the coefficient sequences are restricted to be in some lp space (1 ≦ p ≦ ∞) and we are interested in bounding their lp-norms in terms of the Lp-norm of the linear combination of integer translates of the basis functions which uses these coefficients. In both cases we give necessary and sufficient conditions for linear independence of integer translates of the basis functions. Our characterization is based on a study of certain systems of linear partial difference and differential equations, which are of independent interest.

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1.C. de Boor and K. Höllig , B-splines from parallelepipeds, J. Analyse Math. 42 (1982/1983), 99115.

2.C. de Boor and A. Ron , On polynomial ideals of finite codimension with applications to box spline theory, Math. Anal. & Appl. 158 (1991), 168193.

3.C. K. Chui , Multivariate Splines (CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 54, SIAM, Philadelphia, 1988).

4.W. Dahmen , R. Q. Jia and C. A. Micchelli , On linear dependence relations for integer translates of compactly supported distributions, Math. Nachr. 151 (1991), 303310.

6.W. Dahmen and C. A. Micchelli , On the solution of certain systems of partial difference equations and linear dependence of translates of box splines, Trans. Amer. Math. Soc. 292 (1985), 305320.

7.W. Dahmen and C. A. Micchelli , Local dimension of piecewise polynomial spaces, syzygies, and solutions of systems of partial differential equations, Math. Nachr. 148 (1990), 117136.

8.G. Fix and G. Strang , Fourier analysis of the finite element method in Ritz-Galerkin theory, Stud. Appl. Math. 48 (1969), 265273.

10.R. Q. Jia , Linear independence of translates of a box spline, J. Approx. Theory 40 (1984), 158160.

11.R. Q. Jia , S. Riemenschneider and Z. W. Shen , Dimension of kernels of linear operators, Amer. J. Math. 114 (1992), 157184.

13.A. Ron , A necessary and sufficient condition for the linear independence of the integer translates of a compactly supported distribution, Constr. Approx. 5 (1989), 297308.

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