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A Class of Interpolating Positive Linear Operators: Theoretical and Computational Aspects

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Approximation Theory, Wavelets and Applications

Part of the book series: NATO Science Series ((ASIC,volume 454))

Abstract

This paper reports expository talks, presented at the NATO-ASI, on scattered data interpolation by means of positive linear operators, relating to classical and extended operators of Shepard’s type. Emphasis is placed on some topics such as constructive procedures, convergence and rate of approximation, connection with physical models, computational problems, algorithms for parallel, multistage and recursive computation.

This work has been supported by the Italian Ministry of Scientific and Technological Research and the National Research Council.

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References

  1. Alfeld, P., Scattered data interpolation in three or more variables, in T. Lyche and L. L. Schumaker (eds.), Mathematical methods in computer aided geometric design, Academic Press, Boston, 1989, 1–33.

    Google Scholar 

  2. Allasia, G., Un’analisi del concetto di media: profilo storico-critico, Atti Accad. Sci. Torino, 114 (1980), 111–123.

    MathSciNet  MATH  Google Scholar 

  3. Allasia, G., Some physical and mathematical properties of inverse distance weighted methods for scattered data interpolation, Calcolo 29 1–2 (1992a), 97–109.

    Google Scholar 

  4. Allasia, G., Parallel and recursive computation of Shepard type formulas, Univ. Torino, Dept. Math., Report 1992b.

    Google Scholar 

  5. Allasia, G. and R. Besenghi, Properties of interpolating means with exponential-type weights, to appear in P. J. Laurent, A. Le Méhauté and L.L. Schumaker (eds.), Curves and surfaces II, A. K. Peters, Boston, 1994.

    Google Scholar 

  6. Allasia, G. and R. Besenghi, Multivariable interpolating means with exponential-type weights for scattered data, to appear.

    Google Scholar 

  7. Allasia, G., R. Besenghi and V. Demichelis, Weighted arithmetic means possessing the interpolation property, Calcolo 253 (1988), 203–217.

    Article  MathSciNet  Google Scholar 

  8. Allasia, G., R. Besenghi and V. Demichelis, Multivariable interpolation by weighted arithmetic means at arbitrary points, Calcolo 29 3–4 (1992), 301–311.

    Article  MathSciNet  MATH  Google Scholar 

  9. Baranov, W., Potential fields and their transformations in applied geophysics, Gebrüder Borntraeger, Berlin, 1975, 82–89.

    Google Scholar 

  10. Barnhill, R. E., Representation and approximation of surfaces, in J. R. Rice (ed.), Mathematical software III, Academic Press, New York, 1977, 69–120.

    Google Scholar 

  11. Barnhill, R. E., R. P. Dube and F. F. Little, Properties of Shepard’s surfaces, Rocky Mountain J. Math. 13 2 (1983), 365–382.

    Article  MathSciNet  MATH  Google Scholar 

  12. Barnhill, R. E. and S. E. Stead, Multistage trivariate surfaces, Rocky Mountain J. Math. 14 1 (1984), 103–118.

    Article  MathSciNet  MATH  Google Scholar 

  13. Belli, G., S. Cerlesi, E. Milani and S. Ratti, Statistical interpolation model for the description of ground pollution etc., Toxicological and Environmental Chemistry, 22 (1989), 101–130.

    Article  Google Scholar 

  14. Berezin, I. S. and N. P. Zhidkov, Computing methods, vol. I, Pergamon Press, Oxford 1965, 170–171.

    MATH  Google Scholar 

  15. Boehm, W., G. Farin and J. Kahmann, A survey of curve and surface methods in CAGD, Comput. Aided Geom. Design 1 (1984), 1–60.

    Article  MATH  Google Scholar 

  16. Boone, D. R. and G. S. Samuelson, Computer mapping for air quality, J. Environ. Engrg. Div., ASCE, 103 EE6, (1977), 969–979.

    Google Scholar 

  17. Bos, L. P. and K. Salkauskas, Moving least-squares are Backus-Gilbert optimal, J. Approx. Theory 59 (1989), 267–275.

    Article  MathSciNet  MATH  Google Scholar 

  18. Carlson, R. E. and T. A. Foley, Interpolation of track data with radial basis methods, Computers Math. Applic. 24 12 (1992), 27–34.

    Article  MathSciNet  MATH  Google Scholar 

  19. Censor, E., Quantitative results for positive linear approximation operators, J. Approx. Theory 4 (1971), 442–450.

    Article  MathSciNet  MATH  Google Scholar 

  20. Cheney, E. W., Multivariate approximation theory: selected topics, CBMS-NSF Regional Conference Series in Applied Mathematics Vol. 51, SIAM, Philadelphia, 1986.

    Book  Google Scholar 

  21. Coman, Gh., The remainder of certain Shepard type interpolation formulas, Studia Univ. Babeş Bolyai, Math., XXXII 4 (1987), 25–33.

    MathSciNet  Google Scholar 

  22. Coman, Gh. and L. Ţfâmbulea, A Shepard-Taylor approximation formula, Studia Univ. Babeş Bolyai, Math., XXXIII 3 (1988), 65–71.

    Google Scholar 

  23. Coman, Gh. and L. Ţfâmbulea, On some interpolation procedures of scattered data, Studia Univ. Babeş Bolyai, Math., XXXV 2 (1990), 90–98.

    Google Scholar 

  24. Crain, I. K. and B. K. Bhattacharyya, Treatment of non-equispaced two-dimensional data with a digital computer, Geoexploration 5 (1967), 173–194.

    Article  Google Scholar 

  25. Cressman, G. P., An operational objective analysis system, Monthly Weather Review 87 (1959), 367–374.

    Article  Google Scholar 

  26. Criscuolo, G. and G. Mastroianni, Estimatesof the Shepard interpolatory procedure, Acta Math. Acad. Sci. Hungar. 61 1–2 (1993), 79–91.

    MathSciNet  MATH  Google Scholar 

  27. Criscuolo, G., G. Mastroianni and P. Nevai, Some convergence estimates of a linear positive operator, in C. K. Chui, L. L. Schumaker and J. D. Ward (eds.), Approximation Theory VI, Vol. I, Academic Press, New York, 1989, 153–157.

    Google Scholar 

  28. Della Vecchia, B. and G. Mastroianni, Pointwise simultaneous approximation by rational operators, Journ. Approx. Th. 65 (1991a), 140–150.

    Article  MATH  Google Scholar 

  29. Della Vecchia, B. and G. Mastroianni, Pointwise estimates of rational operators based on general distributions of knots, Facta Universitatis (Niš), Ser. Math. Inform. 6 (1991) 63–72.

    MATH  Google Scholar 

  30. Della Vecchia, B., G. Mastroianni and V. Totik, Saturation of the Shepard operators, Approx. Theory Appl. 6 (1990), 76–84.

    MathSciNet  MATH  Google Scholar 

  31. Erdős, P. and P. Vértesi, On certain saturation problems, Acta Math. Hung. 53 (1989), 197–203.

    Article  Google Scholar 

  32. Farwig, R., Rate of convergence of Shepard’s global interpolation formula, Math. Comp. 46 174 (1986a), 577–590.

    MathSciNet  MATH  Google Scholar 

  33. Farwig, R., Multivariate interpolation of arbitrarily spaced data by moving least squares methods, J. Comp. Appl. Math. 16 (1986b), 79–93.

    Article  MathSciNet  MATH  Google Scholar 

  34. Farwig, R., Multivariate interpolation of scattered data by moving least squares methods, in M. G. Cox and J. C. Mason (eds.), Algorithms for approximation, Clarendon Press, Oxford, 1987, 193–211.

    Google Scholar 

  35. Farwig, R., Rate of convergence of moving least squares interpolation methods: the univariate case, in P. Nevai and A. Pinkus (eds), Progress in approximation theory, Academic Press, Boston, 1991, 313–327

    Google Scholar 

  36. Feller W., An introduction to probability theory and its applications, Vol.II, 2nd ed., Wiley, New York, 1971, 219–246.

    Google Scholar 

  37. Franke, R., Locally determined smooth interpolation at irregularly spaced points in several variables, J. Inst. Math. Appl. 19 (1977), 471–482.

    Article  MathSciNet  MATH  Google Scholar 

  38. Franke, R., A critical comparison of some methods for interpolation of scattered data, Techn. Report NPS-53–79–003, Naval Postgraduate School, Monterey, California, 1979.

    Google Scholar 

  39. Franke, R., Scattered data interpolation: tests of some methods, Math. Comp. 38 157 (1982), 181–200.

    MathSciNet  MATH  Google Scholar 

  40. Franke, R. and G. Nielson, Smooth interpolation of large sets of scattered data, Intern. J. Numer. Methods Engrg. 15 (1980), 1691–1704.

    Article  MathSciNet  MATH  Google Scholar 

  41. Goodin, W. R., G. J. McRae and J. H. Seinfeld, A comparison of interpolation methods for sparse data: application to wind and concentration fields, J. Appl. Meteor. 18 (1979), 761–771.

    Article  Google Scholar 

  42. Gordon, W. J. and J. A. Wixom, Shepard method of “metric interpolation” to bivariate and multivariate interpolation, Math. Comp. 32 141 (1978), 253–264.

    MathSciNet  MATH  Google Scholar 

  43. Hayes, J. C., Nag algorithms for the approximation of functions and data, in M. C. Cox and J. C. Mason (eds.), Algorithms for approximation, Clarendon Press, Oxford, 1987, 653–668.

    Google Scholar 

  44. Heble, M. P., Approximations problems in analysis and probability, North-Holland, Amsterdam, 1989, 169–177.

    Google Scholar 

  45. Hermann, T. and P. Vértesi, On an interpolatory operator and its saturation, Acta Math. Acad. Sci. Hungar. 37 (1981), 1–9.

    Article  MathSciNet  MATH  Google Scholar 

  46. Katkauskayte, A. Y., The rate of convergence of a Shepard estimate in the presence of random interference, Avtomatika 5 (1991), 21–26; transl. Soviet J. Automat. Inform. Sci. 24 5 (1991), 19–24.

    Google Scholar 

  47. King, J. P., Probability and positive linear operators, Rev. Roum. Math. Pures Appl. 20 3 (1975), 325–327.

    MATH  Google Scholar 

  48. Korovkin, P. P., Linear operators and approximation theory, Hindustan Publ. Corp., Delhi, 1960.

    Google Scholar 

  49. Kunz, K. S., Numerical analysis, McGraw-Hill, New York, 1957, 266–267.

    MATH  Google Scholar 

  50. Lancaster, P. and K. Salkauskas, Surfaces generated by moving least squares methods, Math. Comp. 37 155 (1981), 141–158.

    Article  MathSciNet  MATH  Google Scholar 

  51. Little, F. F., Convex combination surfaces, in R. E. Barnhill and W. Boehm (eds.), Surfaces in Computer Aided Geometric Design, NorthHolland, Amsterdam, 1983, 99–107.

    Google Scholar 

  52. McLain, D. H., Drawing contours from arbitrary data points, Computer J. 17 (1974), 318–324.

    Article  Google Scholar 

  53. NAG Fortran Library Manual. Mark 13, NAG Central Office, 7 Banbury Road, Oxford, OX26NN, U.K., 1988.

    Google Scholar 

  54. Newman, D. J. and T. J. Rivlin, Optimal universally stable interpolation, IBM Research Rpt. RC 9751, New York, 1982.

    Google Scholar 

  55. Nielson, G. M., Coordinate free scattered data interpolation, in C. K. Chui, L. L. Schumaker and F. I. Utreras (eds.), Topics in multivariate approximation, Academic Press, 1987, 175–184.

    Google Scholar 

  56. Pelto, C. R., T. A. Elkins and H. A. Boyd, Automatic contouring of irregularly spaced data, Geophysics 33 3 (1968), 424–430.

    Article  Google Scholar 

  57. Prenter, P. M., Lagrange and Hermite interpolation in Banach spaces, J. Approx. Theory 4 (1971), 419–432.

    Article  MathSciNet  MATH  Google Scholar 

  58. Renka, R. J., Multivariate interpolation of large sets of scattered data, ACM Trans. Math. Softw. 14 2 (1988), 139–148.

    Article  MathSciNet  MATH  Google Scholar 

  59. Schneider, C. and W. Werner, Hermite interpolation: the barycentric approach, Computing 46 (1991), 35–51.

    Article  MathSciNet  MATH  Google Scholar 

  60. Schumaker, L. L., Fitting surfaces to scattered data, in G. G. Lorentz, C. K. Chui and L. L. Schumaker (eds.), Approximation Theory II, Academic Press, New York, 1976, 203–268.

    Google Scholar 

  61. Shen, C. Y., H. L. Reed and T. A. Foley, Shepard’s interpolation for solution-adaptive methods, J. Comput. Phys. 106 (1993), 52–61.

    Article  MathSciNet  MATH  Google Scholar 

  62. Shepard D., A two-dimensional interpolation function for irregularly spaced data, Proc. 23rd Nat. Conf. ACM, Brandon/Systems Press Inc., Princeton, 1968a, 517–524.

    Google Scholar 

  63. Shepard D., A two-dimensional interpolation function for computer mapping of irregularly spaced data, Tech. Report ONR-15, March 1968b, Harvard Univ., Cambridge, Mass.

    Google Scholar 

  64. Somorjai, G., On a saturation problem, Acta Math. Acad. Sci. Hungar. 32 (1978), 377–381.

    Article  MathSciNet  MATH  Google Scholar 

  65. Stancu, D. D., Use of probabilistic methods in the theory of uniform approximation of continuous functions, Rev. Roumaine Math. Pures Appl., 14 5 (1969), 673–691.

    MathSciNet  MATH  Google Scholar 

  66. Stancu, D. D., Probabilistic methods in the theory of approximation of functions of several variables by linear positive operators, in A. Talbot (ed.), Approximation Theory, Academic Press, London, 1970, 329–342.

    Google Scholar 

  67. Stancu, D. D., Probabilistic approach to a class of generalized Bernstein approximating operators, Mathematica, Revue d’Analyse Numérique et de la Théorie de l’Approximation, 14 1 (1985), 83–89.

    MathSciNet  MATH  Google Scholar 

  68. Szabados, J., On a problem of R. DeVore, Acta Math. Acad. Sci. Hungar. 27 (1976), 219–223.

    Article  MathSciNet  MATH  Google Scholar 

  69. Szabados, J., Direct and converse approximation theorems for the Shepard operator, Approx. Theory Appl. 7 (1991), 63–76.

    MathSciNet  MATH  Google Scholar 

  70. Wilks, S. S., Mathematical statistics, Wiley, New York, 1962.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123, Torino, Italy

    G. Allasia

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  1. G. Allasia
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Editors and Affiliations

  1. Memorial University of Newfoundland, St. John’s, Newfoundland, Canada

    S. P. Singh

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Allasia, G. (1995). A Class of Interpolating Positive Linear Operators: Theoretical and Computational Aspects. In: Singh, S.P. (eds) Approximation Theory, Wavelets and Applications. NATO Science Series, vol 454. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8577-4_1

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  • DOI: https://doi.org/10.1007/978-94-015-8577-4_1

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4516-4

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