Summary
Let ϕ be a compactly supported function on ℝs andS (ϕ) the linear space withgenerator ϕ; that is,S (ϕ) is the linear span of the multiinteger translates of ϕ. It is well known that corresponding to a generator ϕ there are infinitely many quasi-interpolation formulas. A characterization of these formulas is presented which allows for their direct calculation in a variety of forms suitable to particular applications, and in addition, provides a clear formulation of the difficult problem of minimally supported quasi-interpolants. We introduce a generalization of interpolation called μ-interpolation and a notion of higher order quasi-interpolation called μ-approximation. A characterization of μ-approximants similar to that of quasi-interpolants is obtained with similar applications. Among these applications are estimating least-squares approximants without matrix inversion, surface fitting to incomplete or semi-scattered discrete data, and constructing generators with one-point quasi-interpolation formulas. It will be seen that the exact values of the generator ϕ at the multi-integers ℤs facilitates the above study. An algorithm to yield this information for box splines is discussed.
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References
B[87] Boor, C.: The polynomials in the linear span of integer translates of a compactly supported function. Const. Approx.3, 199–208 (1987)
BJ[85] de Boor, C., Jia, R.Q.: Controlled approximation and a characterization of the local approximation order. Proc. Am. Math. Soc.95, 547–553 (1985)
C[88] Chui, C.K.: Multivariate Splines. CBMS Series in Applied Mathematics, SIAM Publications, Philadelphia, 1988
CD[87] Chui, C.K., Diamond, H.: A natural formulation of quasi-interpolation by multivariate splines. Proc. Am. Math. Soc.99, 643–646 (1987)
CDR[88] Chui, C.K., Diamond, H., Raphael, L.A.: Interpolation by multivariate splines. Math. Comput.51, 203–218 (1988)
CJW[87] Chui, C.K., Jetter, K., Ward, J.: Cardinal interpolation by multivariate splines. Math. Comput.48, 711–724 (1987)
CL[87a] Chui, C.K., Lai, M.J.: Vandermonde determinants and Lagrange interpolations in ℝs. In: Lin, B.L., Simons, S. (eds.). Nonlinear and Convex Analysis, pp. 23–25. New York: Marcel Dekker 1987
CL[87b] Chui, C.K., Lai, M.J.: A multivariate analog of Marsden's identity and a quasi-interpolation scheme. Const. Approx.3, 111–122 (1987)
CY[77] Chung, K.C., Yao, T.H.: On lattices admitting unique Lagrange interpolations. SIAM J. Numer. Anal.14, 735–743 (1977)
DDL[85] Dahmen, W., Dyn, N., Levin, D.: On the convergence of subdivision algorithms for box spline surfaces. Const. Approx.1, 305–322 (1985)
RS[87] Riemenschneider, S., Scherer, K.: Cardinal Hermite interpolation with box splines. Const. Approx.3, 223–238 (1987)
DM[83] Dahmen, W., Micchelli, C.A.: Translates of multivariate splines. Linear Algebra Appl.52/3, 217–234 (1983)
SF[73] Strang, G., Fix, G.: A Fourier analysis of the finite element variational method, C.I.M.E., II Ciclo 1971, Constructive Aspects of Functional Analysis, Geymonat, G. (ed.) 1973, pp. 793–840
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Supported by the National Science Foundation and the U.S. Army Research Office
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Chui, C.K., Diamond, H. A characterization of multivariate quasi-interpolation formulas and its applications. Numer. Math. 57, 105–121 (1990). https://doi.org/10.1007/BF01386401
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DOI: https://doi.org/10.1007/BF01386401