Abstract
A simple method for constructing almost interpolation sets in the case of existence of locally linearly independent systems of basis functions is presented. Various examples of such systems, including translates of box splines and finite-element splines, are considered.
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References
Alfeld P., Schumaker L. L., and Sirvent M., On dimension and existence of local bases for multivariate spline spaces, J. Approx. Theory 70 (1992), 243–264.
Ben-Artzi A. and Ron A., On the integer translates of a compactly supported functions: dual bases and linear projectors, SIAM J. Math. Anal. 21 (1990), 1550–1562.
Bojanov B. D., Hakopian H. A., and Sahakian A. A., Spline Functions and Multivariate Interpolations, Kluwer Academic Publishers, Dordrecht, 1993.
de Boor C., B-form basics, in Geometric Modeling: Algorithms and New Trends (Farin G. E., Ed.), SIAM, Philadelphia, 1987, 131–148.
de Boor C. and Höllig K., B-splines from parallelepipeds, J. Analyse Math. 42 (1982), 99–115.
de Boor C. and Höllig K., Bivariate box splines and smooth pp functions on a three direction mesh, J. Comput. Appl. Math. 9 (1983), 13–28.
de Boor C., Höllig K., and Riemenschneider S., Box Splines, Springer, New York, 1993.
Carnicer J. M. and Pefña J. M., Least supported bases and local linear independence, Numer. Math. 67 (1994), 289–301.
Carnicer J. M. and Peña J. M., Spaces with almost strictly totally positive bases, Math. Nachrichten 169 (1994), 69–79.
Cheney E. W., Multivariate Approximation Theory: Selected Topics, CBMS-SIAM, Philadelphia, 1986.
Chui C. K., Multivariate Splines, CBMS-SIAM, Philadelphia, 1988.
Dahmen W. and Micchelli C. A., Translates of multivariate splines, Linear Algebra Appl. 52/53 (1983), 217–234.
Dahmen W. and Micchelli C. A., On the local linear independence of translates of a box spline, Studia Math. 82 (1985), 243–262.
Dahmen W. and Micchelli C. A., On multivariate E-splines, Advances in Math. 76 (1989), 33–93.
Davydov O. and Sommer M., Interpolation by weak Chebyshev spaces, preprint.
Davydov O., Sommer M., and Strauss H., On almost interpolation by multivariate splines, this volume.
Davydov O., Sommer M., and Strauss H., On almost interpolation and locally linearly independent bases, preprint.
Jia R.-Q., Linear independence of translates of a box spline, J. Approx. Theory 40 (1984), 158–160.
Jia R.-Q., Local linear independence of the translates of a box spline, Constr. Approx. 1 (1985), 175–182.
Ron A., Linear independence of the translates of an exponential box spline, Rocky Mountain J. Math. 22 (1992), 331–351.
Schrijver A., Theory of linear and integer programming, Wiley-Interscience, New York, 1986.
Schumaker L. L., Spline Functions: Basic Theory, Wiley-Interscience, New York, 1981.
Schumaker L. L., On super splines and finite elements, SIAM J. Numer. Anal. 26 (1989), 997–1005.
Sommer M. and Strauss H., Weak Descartes systems in generalized spline spaces, Constr. Approx. 4 (1988), 133–145.
Sommer M. and Strauss H., A condition of Schoenberg-Whitney type for multivariate spline interpolation, Advances in Comp. Math. 5 (1996), 381–397.
Sun Q., A note on the integer translates of a compactly supported distribution on ℝ, Arch. Math. 60 (1993), 359–363.
Ženíšek A., Interpolation polynomials on the triangle, Numer. Math. 15 (1970), 283–296.
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© 1997 Springer Basel AG
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Davydov, O., Sommer, M., Strausβ, H. (1997). Locally Linearly Independent Systems and Almost Interpolation. In: Nürnberger, G., Schmidt, J.W., Walz, G. (eds) Multivariate Approximation and Splines. ISNM International Series of Numerical Mathematics, vol 125. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8871-4_5
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DOI: https://doi.org/10.1007/978-3-0348-8871-4_5
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9808-9
Online ISBN: 978-3-0348-8871-4
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