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Published online by Cambridge University Press:  24 February 2022

Martin Buhmann
Affiliation:
Justus-Liebig-Universität Giessen, Germany
Janin Jäger
Affiliation:
Justus-Liebig-Universität Giessen, Germany
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Quasi-Interpolation , pp. 266 - 273
Publisher: Cambridge University Press
Print publication year: 2022

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References

Abbadi, A., Barrera, D., Ibáñez, M. J., and Sbibih, D. 2010. A general method for constructing quasi-interpolants from B-splines. J. Comput. Appl. Math., 234 , 13241337.Google Scholar
Abbas, M., Majid, A. A., Ismail, A. I. M., and Rashid, A. 2014. Numerical method using cubic trigonometric B-spline technique for nonclassical diffusion problems. Abstract Appl. Anal., 2014 , 849682.Google Scholar
Abel, U., and Ivan, M. 2004. The complete asymptotic expansion for the gamma operators and their left quasi-interpolants. Pages 184–189 of: Bojanov, B. (ed), Constructive Theory of Functions (Varna, 2002). Darba.Google Scholar
Abramowitz, M., and Stegun, I. A. 1972. Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables. Dover Publications.Google Scholar
Allouch, C., Sablonnière, P., and Sbibih, D. 2011. Solving Fredholm integral equations by approximating kernels by spline quasi-interpolants. Numer. Algorithms, 56 , 437453.Google Scholar
Ameur, E. B., Sbibih, D., Almhdie, A., and Léger, C. 2007. New spline quasi-interpolant for fitting 3-D data on the sphere: applications to medical imaging. IEEE Signal Process. Lett., 14 , 333336.CrossRefGoogle Scholar
Ameur, E. B., Barrera, D., Ibáñez, M. J., and Sbibih, D. 2008. Near-best operators based on a C 2 quartic spline on the uniform four-directional mesh. Math. Comput. Simulation, 77 , 151160.Google Scholar
Askey, R. 1975. Some characteristic functions of unimodal distributions. J. Math. Anal. Appl., 50 , 465469.CrossRefGoogle Scholar
Barrera, D., Ibáñez, M. J., Sablonnière, P., and Sbibih, D. 2005a. Near-best quasi-interpolants associated with H-splines on a three-direction mesh. J. Comput. Appl. Math., 183 , 133152.CrossRefGoogle Scholar
Barrera, D., Ibáñez, M. J., Sablonnière, P., and Sbibih, D. 2005b. Near minimally normed spline quasi-interpolants on uniform partitions. J. Comput. Appl. Math., 181 , 211233.CrossRefGoogle Scholar
Barrera, D., Ibáñez, M. J., and Sbibih, D. 2008a. Near-best operators based on a C 2 quartic spline on the uniform four-directional mesh. Math. Comput. Simulation, 77 , 151160.Google Scholar
Barrera, D., Ibáñez, M. J., Sablonnière, P., and Sbibih, D. 2008b. Near-best univariate spline discrete quasi-interpolants on nonuniform partitions. Constr. Approx., 28 , 237251.CrossRefGoogle Scholar
Barrera, D., Ibáñez, M. J., Sablonnière, P., and Sbibih, D. 2010. On near-best discrete quasi-interpolation on a four-directional mesh. J. Comput. Appl. Math., 233 , 14701477.CrossRefGoogle Scholar
Barrera, D., Ibáñez, M. J., and Remogna, S. 2017. On the construction of trivariate near-best quasi-interpolants based on C 2 quartic splines on type-6 tetrahedral partitions. J. Comput. Appl. Math., 311 , 252261.CrossRefGoogle Scholar
Barrera, D., Dagnino, C., Ibáñez, M. J., and Remogna, S. 2018a. Trivariate near-best blending spline quasi-interpolation operators. Numer. Algorithms, 78 , 217241.Google Scholar
Barrera, D., Elmokhtari, F., and Sbibih, D. 2018b. Two methods based on bivariate spline quasi-interpolants for solving Fredholm integral equations. Appl. Numer. Math., 127 , 7894.Google Scholar
Baxter, B. J. C. 1992. The interpolation theory of radial basis functions. PhD thesis, University of Cambridge.Google Scholar
Beatson, R. K., and Dyn, N. 1996. Multiquadric B-splines. J. Approx. Theory, 87 , 124.CrossRefGoogle Scholar
Beatson, R. K., and Powell, M. J. D. 1992a. Univariate interpolation on a regular finite grid by a multiquadric plus a linear polynomial. IMA J. Numer. Anal., 12 , 107133.Google Scholar
Beatson, R. K., and Powell, M. J. D. 1992b. Univariate multiquadric approximation: quasi-interpolation to scattered data. Constr. Approx., 8 , 275288.CrossRefGoogle Scholar
Bernstein, S. 1912. Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités. Commun. Kharkov Math. Soc., 13 , 12.Google Scholar
Bleimann, G., and Stark, E. L. 1981. A trigonometric version of Taylor’s formula. Pages 149–162 of: Ciesielski, Z. (ed), Approximation and Function Spaces (Gdańsk, 1979). North-Holland.Google Scholar
de Boor, C. 1987. The polynomials in the linear span of integer translates of a compactly supported function. Constr. Approx., 3 , 199208.CrossRefGoogle Scholar
de Boor, C. 1990. Splinefunktionen. Birkhäuser.CrossRefGoogle Scholar
de Boor, C. 2001. A Practical Guide to Splines. Applied Mathematical Sciences, vol. 27. Springer.Google Scholar
de Boor, C. 2009. The way things were in multivariate splines: a personal view. Pages 19–37 of: DeVore, R., and Kunoth, A. (eds), Multiscale, Nonlinear and Adaptive Approximation. Springer.Google Scholar
de Boor, C., and Fix, G. J. 1973. Spline approximation by quasiinterpolants. J. Approx. Theory, 8 , 1945.CrossRefGoogle Scholar
de Boor, C., and Ron, A. 1992. Fourier analysis of the approximation power of principal shift-invariant spaces. Constr. Approx., 8 , 427462.CrossRefGoogle Scholar
de Boor, C., Höllig, K., and Riemenschneider, S. 1993a. Box Splines. Springer.Google Scholar
de Boor, C., DeVore, R. A., and Ron, A. 1993b. On the construction of multivariate (pre)wavelets. Constr. Approx., 9 , 123166.Google Scholar
de Boor, C., DeVore, R. A., and Ron, A. 1994. Approximation from shift-invariant subspaces of L 2 (R d ). Trans. Amer. Math. Soc., 341 , 787806.Google Scholar
Broomhead, D. S., and Lowe, D. 1988. Radial basis functions, multi-variable functional interpolation and adaptive networks. Technical report, Royal Signals and Radar Establishment, Malvern, UK.Google Scholar
Buffa, A., Garau, E. M., Giannelli, C., and Sangalli, G. 2016. On quasi-interpolation operators in spline spaces. Pages 7391 of: Barrenechea, G., et al. (eds), Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol. 114. Springer.Google Scholar
Buhmann, M. D. 1988. Convergence of univariate quasi-interpolation using multiquadrics. IMA J. Numer. Anal., 8 , 365383.Google Scholar
Buhmann, M. D. 1989. Multivariable interpolation using radial basis functions. PhD thesis, University of Cambridge.Google Scholar
Buhmann, M. D. 1990a. Multivariate cardinal interpolation with radial-basis functions. Constr. Approx., 6 , 225255.CrossRefGoogle Scholar
Buhmann, M. D. 1990b. Multivariate interpolation in odd-dimensional Euclidean spaces using multiquadrics. Constr. Approx., 6 , 2134.Google Scholar
Buhmann, M. D. 1993a. Discrete least squares approximation and prewavelets from radial function spaces. Math. Proc. Cambridge Philos. Soc., 114 , 533558.Google Scholar
Buhmann, M. D. 1993b. On quasi-interpolation with radial basis functions. J. Approx. Theory, 72 , 103130.CrossRefGoogle Scholar
Buhmann, M. D. 2003. Radial Basis Functions: Theory and Implementations. Cambridge University Press.Google Scholar
Buhmann, M. D., and Dai, F. 2015a. Compression using quasi-interpolation. Jaen J. Approx., 7 , 203230.Google Scholar
Buhmann, M. D., and Dai, F. 2015b. Pointwise approximation with quasi-interpolation by radial basis functions. J. Approx. Theory, 192 , 156192.Google Scholar
Buhmann, M. D., and Dinew, S. 2007. Limits of radial basis function interpolants. Commun. Pure Appl. Anal., 6 , 569585.Google Scholar
Buhmann, M. D., and Dyn, N. 1991. Error estimates for multiquadric interpolation. Pages 51–58 of: Laurent, P.-J., et al. (eds), Curves and Surfaces (Chamonix–Mont-Blanc, 1990). Academic Press.CrossRefGoogle Scholar
Buhmann, M. D., and Dyn, N. 1993. Spectral convergence of multiquadric interpolation. Proc. Edinb. Math. Soc., 36 , 319333.CrossRefGoogle Scholar
Buhmann, M. D., and Jäger, J. 2020a. Multiply monotone functions for radial basis function interpolation: extensions and new kernels. J. Approx. Theory, 256 , 105434.Google Scholar
Buhmann, M. D., and Jäger, J. 2020b. Pólya type criteria for conditional strict positive definiteness of functions on spheres. J. Approx. Theory, 257 , 105440.Google Scholar
Buhmann, M. D., and Jäger, J. 2021. Strictly positive definite kernels on the 2-sphere: from radial symmetry to eigenvalue block structure. IMA J . Numer. Analysis . https://doi.org/10.1093/imanum/drab012 Google Scholar
Buhmann, M. D., and Micchelli, C. A. 1991. Multiply monotone functions for cardinal interpolation. Adv. Appl. Math., 12 , 358386.Google Scholar
Buhmann, M. D., and Micchelli, C. A. 1992a. Multiquadric interpolation improved. Comput. Math. Appl., 24, 21–25. Special issue: Advances in the Theory and Applications of Radial Basis Functions.CrossRefGoogle Scholar
Buhmann, M. D., and Micchelli, C. A. 1992b. On radial basis approximation on periodic grids. Math. Proc. Cambridge Philos. Soc., 112 , 317334.CrossRefGoogle Scholar
Buhmann, M. D., and Micchelli, C. A. 1992c. Spline prewavelets for nonuniform knots. Numer. Math., 61 , 455474.Google Scholar
Buhmann, M. D., Dai, F., and Niu, Y. 2021. Discretization of integrals on compact metric measure spaces. Advances in Mathematics. 107602.Google Scholar
Buhmann, M. D., Dyn, N., and Levin, D. 1995. On quasi-interpolation by radial basis functions with scattered centers. Constr. Approx., 11 , 239254.CrossRefGoogle Scholar
Buhmann, M. D., Dinew, S., and Larsson, E. 2010. A note on radial basis function interpolant limits. IMA J. Numer. Anal., 30 , 543554.CrossRefGoogle Scholar
Buhmann, M. D., Jodar, J., and Rodriguez, M. L. 2019. Radial discrete PDE splines on Lipschitz domains. J. Math. Anal. Appl., 479 , 214241.Google Scholar
Butzer, P. L., and Nessel, R. J. 1971. Fourier Analysis and Approximation , vol. 1. Birkhäuser.Google Scholar
Butzer, P. L., Schmidt, M., Stark, E. L., and Vogt, L. 1989. Central factorial numbers: their main properties and some applications. Numer. Funct. Anal. Optim., 10 , 419488.CrossRefGoogle Scholar
Chen, Z., and Cao, F. 2015. Spherical scattered data quasi-interpolation by Gaussian radial basis function. Chin. Ann. Math. Ser. B, 36 , 401412.CrossRefGoogle Scholar
Chen, Z., Cao, F., and Li, M. 2015. Scattered data quasi-interpolation on spheres. Math. Methods Appl. Sci., 38 , 25272536.Google Scholar
Cheney, W., and Light, W. 2009. A Course in Approximation Theory. Graduate Studies in Mathematics, vol. 101. American Mathematical Society.CrossRefGoogle Scholar
Cheung, K.C., Ling, L.L, and Schaback, R. 2018. H 2-convergence of least squares kernel collocation methods, SIAM J. Numer. Analysis, 56 , 614633.Google Scholar
Chui, C. K. 1988. Multivariate Splines. CBMS Lecture Notes, vol. 54. Society for Industrial and Applied Mathematics.Google Scholar
Chui, C. K., and Diamond, H. 1987. A natural formulation of quasi-interpolation by multivariate splines. Proc. Amer. Math. Soc., 99 , 643646.Google Scholar
Chui, C. K., and Lai, M. J. 1987. A multivariate analog of Marsden’s identity and a quasi-interpolation scheme. Constr. Approx., 3 , 111122.Google Scholar
Chui, C. K., and Wang, J.-Z. 1992. On compactly supported spline wavelets and a duality principle. Trans. Amer. Math. Soc., 330 , 903915.Google Scholar
Chui, C. K., Jetter, K., and Ward, J. D. 1987. Cardinal interpolation by multivariate splines. Math. Comp., 48 , 711724.CrossRefGoogle Scholar
Ciarlet, P. G. 2002. The Finite Element Method for Elliptic Problems. Society of Industrial and Applied Mathematics.Google Scholar
Costantini, P., Manni, C., Pelosi, F., and Sampoli, M. L. 2010. Quasi-interpolation in isogeometric analysis based on generalized B-splines. Comput. Aided Geom. Design, 27 , 656668.Google Scholar
Dagnino, C., Remogna, S., and Sablonnière, P. 2014. On the solution of Fredholm integral equations based on spline quasi-interpolating projectors. BIT Numer. Math., 54 , 979– 1008.Google Scholar
Dagnino, C., Lamberti, P., and Remogna, S. 2015. Near-best C 2 quartic spline quasi-interpolants on type-6 tetrahedral partitions of bounded domains. Calcolo, 52 , 475– 494.CrossRefGoogle Scholar
Dahmen, W., and Micchelli, C. A. 1983. Translates of multivarlate splines. Linear Algebra Appl., 52 , 217234.Google Scholar
Dahmen, W., and Micchelli, C. A. 1984. On the approximation order from certain multivariate spline spaces. J . Aust. Math. Soc. Ser. B Appl. Math., 26 , 233246.Google Scholar
Derriennic, M. M. 1995. de la Vallée Poussin and Bernstein-type operators. Pages 71–84 of: Approximation Theory (Witten, 1995). Mathematical Research, vol. 86. Akademie-Verlag.Google Scholar
DeVore, R. A., and Lorentz, G. G. 1993. Constructive Approximation. Springer.Google Scholar
Dyn, N. 1989. Interpolation and approximation by radial and related functions. Pages 211– 234 of: Chui, C. K., et al. (eds), Approximation Theory VI, vol. 1. Academic Press.Google Scholar
Forsey, D., and Bartels, R. H. 1988. Hierarchical B-spline refinement. ACM Siggraph Computer Graphics, 22 , 205212.Google Scholar
Franke, C., and Schaback, R. 1998. Solving partial differential equations by collocation using radial basis functions. Appl. Math. Comput., 93 , 7382.Google Scholar
Ganesh, M., and Mhaskar, H. N. 2006. Quadrature-free quasi-interpolation on the sphere. Electron. Trans. Numer. Anal., 25 , 101114.Google Scholar
Gao, W., and Wu, Z. 2014. A quasi-interpolation scheme for periodic data based on multiquadric trigonometric B-splines. J. Comput. Appl. Math., 271 , 2030.Google Scholar
Gao, W., and Wu, Z. 2015a. Approximation orders and shape preserving properties of the multiquadric trigonometric B-spline quasi-interpolant. Comput. Math. Appl., 69 , 696707.Google Scholar
Gao, W., and Wu, Z. 2015b. Solving time-dependent differential equations by multiquadric trigonometric quasi-interpolation. Appl. Math. Comput., 253 , 377386.Google Scholar
Gao, W., and Zhang, R. 2018. Multiquadric trigonometric spline quasi-interpolation for numerical differentiation of noisy data: a stochastic perspective. Numer. Algorithms, 77 , 243259.CrossRefGoogle Scholar
Gao, W.-W., and Wang, Z.-G. 2014. A meshless scheme for partial differential equations based on multiquadric trigonometric B-spline quasi-interpolation. Chin. Phys. B, 23 , 110207.Google Scholar
Gautschi, W. 2004. Orthogonal Polynomials: Computation and Approximation. Numerical Mathematics and Scientific Computation. Oxford University Press.Google Scholar
Gel’fand, I. M., and Shilov, G. E. 1964. Generalized Functions , vol. 1, Properties and Operations . Academic Press.Google Scholar
Giannelli, C., Jüttler, B., and Speleers, H. 2012. THB-splines: the truncated basis for hierarchical splines. Comput. Aided Geom. Design, 29 , 485498.Google Scholar
Gomes, S. M., Kushpel, A. K., and Levesley, J. 2001. Approximation in L 2 Sobolev spaces on the 2-sphere by quasi-interpolation. J. Fourier Anal. Appl., 7 , 283295.Google Scholar
Gonsor, D., and Neamtu, M. 1996. Null spaces of differential operators, polar forms, and splines. J. Approx. Theory, 86 , 81107.Google Scholar
Gradshteyn, I. S., and Ryzhik, I. M. 2014. Table of Integrals, Series, and Products. Academic Press.Google Scholar
Hairer, M. 2014. A theory of regularity structures. Inventio Math., 198 , 269504.CrossRefGoogle Scholar
Hakopian, H. 1983. Integral remainder formula of the tensor product interpolation. Bull. Polish Acad. Sci. Math., 31 , 136144.Google Scholar
Hardy, G. H., Littlewood, J. E., and Pólya, G. 1934. Inequalities. Cambridge University Press.Google Scholar
Hofmann, R. 2013. Neue Verfahren zur Approximation mit radialen Basisfunktionen und numerische Untersuchung der Dagum-Funktionen. PhD thesis, Universität Giessen.Google Scholar
Höllig, K., and Hörner, J. 2013. Approximation and Modelling with Splines. Society for Industrial and Applied Mathematics.Google Scholar
Ibáñez, M. J., Lamnii, A., Mraoui, H., and Sbibih, D. 2010. Construction of spherical spline quasi-interpolants based on blossoming. J. Comput. Appl. Math., 234 , 131145.Google Scholar
Jackson, D. 1930. The Theory of Approximation. American Mathematical Society.Google Scholar
Jackson, I. R. H. 1988. Radial basis function methods of multivariate approximation. PhD thesis, University of Cambridge.Google Scholar
Jackson, I. R. H. 1989. An order of convergence for some radial basis functions. IMA J. Numer. Anal., 9 , 567587.Google Scholar
Jäger, J., Buhmann, M. D., Klein, A., and Skrandies, W. 2016. Reconstruction of electroencephalographic data using radial basis functions. Electroencephalogr. Clin. Neurophysiol., 127 , 19781983.Google ScholarPubMed
Jones, D. S. 1982. The Theory of Generalised Functions. Cambridge University Press.Google Scholar
Jorgenson, J., and Lang, S. 2001. The ubiquitous heat kernel. Pages 655683 of: Engquist, B., and Schmid, W. (eds), Mathematics Unlimited: 2001 and Beyond. Springer.Google Scholar
Kansa, E. J. 1990. Multiquadrics: a scattered data approximation scheme with applications to computational fluid-dynamics, part II, solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput. Math. Appl., 19 , 147161.CrossRefGoogle Scholar
Karlin, S. 1968. Total Positivity , vol. 1. Stanford University Press.Google Scholar
Körner, T. W. 1988. Fourier Analysis. Cambridge University Press.CrossRefGoogle Scholar
Kraft, R. 1997. Adaptive and linearly independent multilevel B-splines. Pages 209–218 of: Surface Fitting and Multiresolution Methods (Chamonix–Mont-Blanc, 1996). Vanderbilt University Press.Google Scholar
Krylov, V. I. 1962. Approximate Calculation of Integrals. Macmillan.Google Scholar
Kushpel, A. K., and Levesley, J. 2000. Quasi-interpolation on the 2-sphere using radial polynomials. J. Approx. Theory, 102 , 141154.Google Scholar
Lamnii, A., Mraoui, H., and Sbibih, D. 2009. Quadratic spherical spline quasi-interpolants on Powell–Sabin partitions. Appl. Numer. Math., 59 , 10941109.Google Scholar
Larsson, E., and Fornberg, B. 2005. Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions. Comput. Math. Appl., 49 , 103130.Google Scholar
Lee, B.-G., Lyche, T., and Mørken, K. 2001. Some examples of quasi-interpolants constructed from local spline projectors. Pages 243252 of: Lyche, T., and Schumaker, L. L. (eds), Innovations in Applied Mathematics. Vanderbilt University Press.Google Scholar
Light, W. A., and Cheney, E. W. 1992. Quasi-interpolation with translates of a function having noncompact support. Constr. Approx., 8 , 3548.Google Scholar
Lyche, T., and Morken, K. 2008. Spline methods draft. Department of Informatics, Center of Mathematics for Applications, University of Oslo.Google Scholar
Lyche, T., and Schumaker, L. L. 1975. Local spline approximation methods. J. Approx. Theory, 15 , 294325.Google Scholar
Lyche, T., and Schumaker, L. L. 2000. A multiresolution tensor spline method for fitting functions on the sphere. SIAM J. Sci Comput., 22 , 724746.Google Scholar
Lyche, T., and Winther, R. 1979. A stable recurrence relation for trigonometric B-splines. J. Approx. Theory, 25 , 266279.Google Scholar
Lyche, T., Schumaker, L. L., and Stanley, S. 1998. Quasi-interpolants based on trigonometric splines. J. Approx. Theory, 95 , 280309.Google Scholar
Lyche, T., Manni, C., and Sablonnière, P. 2008. Quasi-interpolation projectors for box splines. J. Comput. Appl. Math., 221, 416–429. Special issue: Recent Progress in Spline and Wavelet Approximation.Google Scholar
Madych, W., and Nelson, S. 1990. Polyharmonic cardinal splines. J. Approx. Th., 60 , 141156.Google Scholar
Maz’ya, V., and Schmidt, G. 1996. On approximate approximations using Gaussian kernels. IMA J. Numer. Anal., 16 , 1329.Google Scholar
Maz’ya, V., and Schmidt, G. 2001. On quasi-interpolation with non-uniformly distributed centers on domains and manifolds. J. Approx. Theory, 110 , 125145.Google Scholar
Maz’ya, V., and Schmidt, G. 2007. Approximate Approximations. Mathematical Surveys and Monographs, vol. 141. American Mathematical Society.Google Scholar
Meyer, Y. 1992. Wavelets and Operators. Cambridge Studies in Advanced Mathematics, vol. 37. Cambridge University Press.Google Scholar
Mhaskar, H., Narcowich, F., and Ward, J. 2001. Spherical Marcinkiewicz–Zygmund inequalities and positive quadrature. Math. Comp., 70 , 11131130.Google Scholar
Mhaskar, H. N., Narcowich, F. J., and Ward, J. D. 1999. Approximation properties of zonal function networks using scattered data on the sphere. Adv. Comput. Math., 11 , 121137.Google Scholar
Micchelli, C. A. 1986. Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constr. Approx., 2 , 1122.Google Scholar
Müller, C. 1966. Spherical Harmonics. Lecture Notes in Mathematics, vol. 17. Springer.Google Scholar
Nazir, T., Abbas, M., Ismail, A. I. M., Majid, A. A., and Rashid, A. 2016. The numerical solution of advection–diffusion problems using new cubic trigonometric B-splines approach. Appl. Math. Model., 40 , 45864611.Google Scholar
Nouisser, O., Sbibih, D., and Sablonnière, P. 2003. A family of spline quasi-interpolants on the sphere. Numer. Algorithms, 33 , 399413.CrossRefGoogle Scholar
Pollandt, R. 1997. Solving nonlinear differential equations of mechanics with the boundary element method and radial basis functions. Internat. J. Numer. Methods Engrg, 40 , 6173.Google Scholar
Porcu, E., Daley, D. J., Buhmann, M. D., and Bevilacqua, M. 2013. Radial basis functions with compact support for multivariate geostatistics. Stoch. Environ. Res. Risk Assess., 27 , 909922.Google Scholar
Porcu, E., Zastavnyi, V. P., and Bevilacqua, M. 2016. Buhmann covariance functions, their compact supports, and their smoothness. Available at arXiv:1606.09527.Google Scholar
Powell, M. J. D. 1981. Approximation Theory and Methods. Cambridge University Press.Google Scholar
Powell, M. J. D. 1990. Univariate multiquadric approximation: reproduction of linear polynomials. Pages 227–240 of: Multivariate Approximation and Interpolation (Duisburg, 1989). International Series of Numerical Mathematics, vol. 94. Birkhäuser.Google Scholar
Powell, M. J. D. 1991. Univariate multiquadric interpolation: some recent results. Pages 371–382 of: Laurent, P.-J., et al. (eds), Curves and Surfaces (Chamonix–Mont-Blanc, 1990). Academic Press.Google Scholar
Rabut, C. 1992. An introduction to Schoenberg’s approximation. Comput. Math. Appl., 24 , 149175.Google Scholar
Remogna, S. 2011. Quasi-interpolation operators based on the trivariate seven-direction C 2 quartic box spline. BIT Numer. Math., 51 , 757776.Google Scholar
Remogna, S., and Sablonnière, P. 2011. On trivariate blending sums of univariate and bivariate quadratic spline quasi-interpolants on bounded domains. Comput. Aided Geom. Design, 28 , 89.Google Scholar
Ron, A. 1988. Exponential box splines. Constr. Approx., 4 , 357378.Google Scholar
Ron, A. 1992. The L 2-approximation orders of principal shift-invariant spaces generated by a radial basis function. Pages 245–268 of: Numerical Methods in Approximation Theory Vol. 9 (Oberwolfach, 1991). International Series of Numerical Mathematics, vol. 105. Birkhäuser.Google Scholar
Sablonnière, P. 1989. Bernstein quasi-interpolants on [0, 1]. Pages 287–294 of: Multivariate Approximation Theory IV. International Series of Numerical Mathematics, vol. 90. Birkhäuser.Google Scholar
Sablonnière, P. 1992. A family of Bernstein quasi-interpolants on [0,1]. Approx. Theory Appl., 8 , 6276.Google Scholar
Sablonnière, P. 1998. Quasi-interpolants associated with B-splines on the uniform four-directional mesh of the plane. In: Workshop on Multivariate Approximation and Interpolation with Applications in CAGD, Signal and Image Processing.Google Scholar
Sablonnière, P. 1999. Representation of quasi-interpolants as differential operators and applications. Pages 233–253 of: Müller, M. W., et al. (eds), New Developments in Approximation Theory (IDoMAT 98). International Series of Numerical Mathematics, vol. 132. Birkhäuser.Google Scholar
Sablonnière, P. 2002. H-splines and quasi-interpolants on a three directional mesh. Pages 187–201 of: Buhmann, M. D., and Mache, D. H. (eds), Advanced Problems in Constructive Approximation. International Series of Numerical Mathematics, vol. 142. Birkhäuser.Google Scholar
Sablonnière, P. 2005a. Recent progress on univariate and multivariate polynomial and spline quasi-interpolants. Pages 229–245 of: Mache, D. H., et al. (eds), Trends and Applications in Constructive Approximation. International Series of Numerical Mathematics, vol. 151. Birkhäuser.Google Scholar
Sablonnière, P. 2005b. Univariate spline quasi-interpolants and applications to numerical analysis. Rend. Semin. Mat., 63 , 211222.Google Scholar
Sablonnière, P. 2007. Quasi-interpolants splines: exemples et applications. Pages 195–207 of: ESAIM: Proceedings, vol. 20. EDP Sciences.Google Scholar
Sablonnière, P. 2011. Bernstein quasi-interpolants on triangles. Stud. Univ. Babeş-Bolyai Math., 56 , 567585.Google Scholar
Sablonnière, P. 2014. Weierstrass quasi-interpolants. J. Approx. Theory, 180 , 3248.Google Scholar
Sablonnière, P., and Sbibih, D. 1994. Spline integral operators exact on polynomials. Approx. Theory Appl., 10 , 5673.Google Scholar
Schaback, R. 1999. Native Hilbert spaces for radial basis functions, part I. Pages 255–282 of: Müller, M. W., et al. (eds), New Developments in Approximation Theory (IDoMAT 98). International Series of Numerical Mathematics, vol. 132. Birkhäuser.CrossRefGoogle Scholar
Schoenberg, I. J. 1946a. Contributions to the problem of approximation of equidistant data, part A. Quart. Appl. Math., 4 , 4598.Google Scholar
Schoenberg, I. J. 1946b. Contributions to the problem of approximation of equidistant data, part B. Quart. Appl. Math., 4 , 112141.Google Scholar
Schoenberg, I. J. 1964. On trigonometric spline interpolation. J. Math. Mech., 13 , 795825.Google Scholar
Schoenberg, I. J., and Whitney, A. 1953. On Pólya frequence functions, part III, The positivity of translation determinants with an application to the interpolation problem by spline curves. Trans. Amer. Math. Soc., 74 , 246259.Google Scholar
Schumaker, L. L. 1981. Spline Functions: Basic Theory. Krieger.Google Scholar
Schumaker, L. L. 2015. Spline Functions: Computational Methods. Society for Industrial and Applied Mathematics.Google Scholar
Schumaker, L. L., and Traas, C. 1991. Fitting scattered data on spherelike surfaces using tensor products of trigonometric and polynomial splines. Numer. Math., 60 , 133144.Google Scholar
Sorokina, T., and Zeilfelder, F. 2005. Optimal quasi-interpolation by quadratic C 1-splines on type-2 triangulations. Pages 423–438 of: Chui, C. K., et al. (eds), Approximation Theory XI: Gatlinburg 2004. Nashboro Press.Google Scholar
Sorokina, T., and Zeilfelder, F. 2007. Local quasi-interpolation by cubic C 1 splines on type-6 tetrahedral partitions. IMA J. Numer. Anal., 27 , 74101.Google Scholar
Sorokina, T., and Zeilfelder, F. 2008. An explicit quasi-interpolation scheme based on C 1 quartic splines on type-1 triangulations. Comput. Aided Geom. Design, 25 , 113.Google Scholar
Speleers, H. 2017. Hierarchical spline spaces: quasi-interpolants and local approximation estimates. Adv. Comput. Math., 43 , 235255.Google Scholar
Speleers, H., and Manni, C. 2016. Effortless quasi-interpolation in hierarchical spaces. Numer. Math., 132 , 155184.Google Scholar
Stein, E. M. 1993. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, vol. 43. Princeton University Press.Google Scholar
Stein, E. M., and Weiss, G. 1971. Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series, vol. 32. Princeton University Press.Google Scholar
Strang, G., and Fix, G. J. 1973. An Analysis of the Finite Element Method. Prentice Hall.Google Scholar
Wahba, G., and Wendelberger, J. 1980. Some new mathematical methods for variational objective analysis using splines and cross validation. Mon. Weather Rev., 108 , 1122– 1143.Google Scholar
Walz, G. 1997. Identities for trigonometric B-splines with an application to curve design. BIT Numer. Math., 37 , 189201.Google Scholar
Wendland, H. 2005. Scattered Data Approximation. Cambridge University Press.Google Scholar
Zastavnyi, V. P. 2006. On some properties of Buhmann functions. Ukrainian J. Math., 58 , 11841208.Google Scholar
Zastavnyi, V. P., and Porcu, E. 2017. On positive definiteness of some radial functions. Lobachevskii J. Math., 38 , 386394.Google Scholar

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  • References
  • Martin Buhmann, Justus-Liebig-Universität Giessen, Germany, Janin Jäger, Justus-Liebig-Universität Giessen, Germany
  • Book: Quasi-Interpolation
  • Online publication: 24 February 2022
  • Chapter DOI: https://doi.org/10.1017/9781139680523.012
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  • References
  • Martin Buhmann, Justus-Liebig-Universität Giessen, Germany, Janin Jäger, Justus-Liebig-Universität Giessen, Germany
  • Book: Quasi-Interpolation
  • Online publication: 24 February 2022
  • Chapter DOI: https://doi.org/10.1017/9781139680523.012
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • References
  • Martin Buhmann, Justus-Liebig-Universität Giessen, Germany, Janin Jäger, Justus-Liebig-Universität Giessen, Germany
  • Book: Quasi-Interpolation
  • Online publication: 24 February 2022
  • Chapter DOI: https://doi.org/10.1017/9781139680523.012
Available formats
×