Abstract
By providing a counterexample we show that there exists a shift-invariant spaceS generated by a piecewise linear function such that the union of the corresponding scaled spacesS h (h>0) is dense inC 0(R 2) butS does not contain a stable and locally supported partition of unity. This settles a question raised by C. de Boor and R. DeVore a decade ago.
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C. de Boor, R. DeVore (1985):Partitions of unity and approximation. Proc. Amer. Math. Soc.,93:705–709.
C. de Boor, R. DeVore, K. Höllig (1983):Approximation order from smooth pp functions. In: Approximation Theory IV (C. K. Chui, L. L. Schumaker, J. Ward, eds.). New York: Academic Press, pp. 353–357.
L. Ehrenpreis (1970): Fourier Analysis in Several Complex Variables. Tracts in Mathematics, vol. 17. New York: Wiley-Interscience.
R.-Q. Jia (1991):A characterization of the approximation order of translation invariant spaces. Proc. Amer. Math. Soc.,111:61–70.
R.-Q. Jia (1995):The Toeplitz theorem and its applications to approximation theory and linear PDEs. Trans. Amer. Math. Soc.,347:2585–2594.
R.-Q. Jia (to appear):Shift-invariant spaces on the real line. Proc. Amer. Math. Soc.
R.-Q. Jia, S. D. Riemenschneider, Z. W. Shen (1994):Solvability of systems of linear operator equations. Proc. Amer. Math. Soc.,120:815–824.
A. Ron (1991):A characterization of the approximation order of multivariate spline spaces. Studia Math.,98:73–90.
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Communicated by Ronald A. DeVore.
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Jia, RQ. Partition of unity and density: A counterexample. Constr. Approx 13, 251–260 (1997). https://doi.org/10.1007/BF02678467
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DOI: https://doi.org/10.1007/BF02678467