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Multidimensional Iterative Interpolation

Published online by Cambridge University Press:  20 November 2018

Gilles Deslauriers ,
Jacques Dubois  and
Serge Dubuc
Gilles Deslauriers
Affiliation:
Département de mathématiques appliquées École Polytechnique, C.P. 6079, Succ. A Montréal, Québec, H3C 3A7
Jacques Dubois
Affiliation:
Département de mathématiques et d'informatique Université de Sherbrooke Sherbrooke, Québec, J1K2R1
Serge Dubuc
Affiliation:
Département de mathématiques et de statistique Université de Montréal, C.P. 6128, Succ. A, Montréal, Québec, H3C 3J7
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Abstract

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We define an iterative interpolation process for data spread over a closed discrete subgroup of the Euclidean space. We describe the main algebraic properties of this process. This interpolation process, under very weak assumptions, is always convergent in the sense of Schwartz distributions. We find also a convenient necessary and sufficient condition for continuity of each interpolation function of a given iterative interpolation process.

Keywords

Iterative interpolation processfundamental interpolating functioncharacteristic functioncontinuous interpolationtemperate distributionFourier transform65D0565D1026B0565Q0542A0542A38
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 2 , 01 April 1991 , pp. 297 - 312
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Daubechies, I. and Lagarias, J., Two-scale Difference Equations I Global Regularity of Solutions & II Local Regularity, Infinite Products of Matrices and Fractals, SIAM J. Math. Anal., (to appear).Google Scholar
2. Deslauriers, G., Dubuc, S., Interpolation dyadique.ln Fractals: Dimensions non entières et applications, Masson Paris, 1987, 44–55.Google Scholar
3. Deslauriers, G., Dubuc, S., Transformées de Fourier de courbes irrégulières, (Comptes Rendus de l'atelier de géométrie fractale. Mai 1986.) Ann. se. math. Québec. (1)11(1987), 25–44.Google Scholar
4. Deslauriers, G., Dubuc, S., Symmetric Iterative Interpolation Processes, Constructive Approximation 5(1989), 49–68.Google Scholar
5. Dubuc, S., Interpolation through an Iterative Scheme, J. of Math. Anal, and Appl.,( 1)114(1986), 185–204.Google Scholar
6. Dyn, N., Levin, D. and Gregory, J.A.,A 4-point interpolatory subdivision scheme for curve design, Computer Aided Geometric Design 4(1987), 257–268.Google Scholar
7. Dyn, N., Levin, D. and Gregory, J.A., Analysis of uniform binary subdivision schemes for curve design, Constructive Approximation, (to appear).Google Scholar
8. Falconer, K.J., The Hausdorff dimension of se If-affine fractals, Math. Proc. Cambridge Philos. Soc. 103(1988), 339–350.Google Scholar
9. Henrici, P., Elements of Numerical Analysis. John Wiley & Sons Inc.,New York, 1964.Google Scholar
10. Kôno, N., On self-affine functions, Japan J. Appl. Math. 3(1986), 259–269.Google Scholar
11. Mandelbrot, B.B., The Fractal Geometry of Nature. W.H. Freeman, San Francisco, 1982.Google Scholar
12. Micchelli, C.A. and Prautzsch, H., Computing surfaces invariant under subdivision, Computer Aided Geometric Design 4(1987), 321–328.Google Scholar
13. Micchelli, C.A. and Prautzsch, H., Uniform refinement of curves, Linear Algebra and Applications, 114/115(1989), 841–870.Google Scholar
14. Smith, G.B., A Fast Surface Interpolation Technique, in Proc. DARPA, Image Understanding Workshop, Oct. 1984,211–215.Google Scholar