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Estimates for Local Approximations of Functions on Differential Manifold

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We obtain an estimate for the convergence rate of approximation of functions on a differentiable manifold We consider two approaches to approximation of such functions. The first is based on approximate relations for the manifold and is independent of a given set of plane approximations. The second is based on approximations of functions in the plane. We consider the approximation of Courant type on the n-dimensional sphere, in the projective space, and also for the Zlamal approximation and interpolation splines.

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References

  1. Yu. K. Dem’yanovich, “Local approximations on manifolds and weighted estimates,” J. Math. Sci., New York 36, No. 2, 261–269 (1987).

  2. Yu. K. Dem’yanovich, Approximation on a Manifold and Minimal Splines [in Russian], St. Petersb. Univ. Press, St. Petersburg (1994).

  3. J. A. Costa and A. O. Hero, “Geodesic entropic graphs for dimension and entropy estimation in manifold learning,” IEEE Trans. Signal Process. 52, No. 8, 2210–2221 (2004).

    Article  MathSciNet  Google Scholar 

  4. Yu. K. Demyanovich, “Spline approximations on manifolds,” Int. J. Wavelets Multiresolut. Inf. Process. 4, No. 3, 383–403 (2006).

    Article  MathSciNet  Google Scholar 

  5. M.-Y. Cheng and H.-t. Wu, “Local linear regression on manifolds and its geometric interpretation,” J. Am. Stat. Assoc. 108, No. 504, 1421–1434 (2013).

    Article  MathSciNet  Google Scholar 

  6. B. Sober and D. Levin, “Manifold approximation by moving least-squares projection (MMLS),” Constr. Approx. 52, No. 3, 443–478 (2020).

    Article  MathSciNet  Google Scholar 

  7. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam etc. (1978).

    MATH  Google Scholar 

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Authors and Affiliations

  1. St. Petersburg State University, 28, Universitetskii pr., Petrodvorets, St. Petersburg, 198504, Russia

    Yu. K. Dem’yanovich

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  1. Yu. K. Dem’yanovich
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Correspondence to Yu. K. Dem’yanovich.

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Translated from Problemy Matematicheskogo Analiza 111, 2021, pp. 67-89.

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Dem’yanovich, Y.K. Estimates for Local Approximations of Functions on Differential Manifold. J Math Sci 257, 624–651 (2021). https://doi.org/10.1007/s10958-021-05514-z

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  • DOI: https://doi.org/10.1007/s10958-021-05514-z

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