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B-Spline Quasi-Interpolation Sampling Representation and Sampling Recovery in Sobolev Spaces of Mixed Smoothness

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Abstract

We proved direct and inverse theorems on B-spline quasi-interpolation sampling representation with a Littlewood-Paley-type norm equivalence in Sobolev spaces \({W^{r}_{p}}\) of mixed smoothness r. Based on this representation, we established estimates of the approximation error of recovery in L q -norm of functions from the unit ball \({U^{r}_{p}}\) in the spaces \({W^{r}_{p}}\) by linear sampling algorithms and the asymptotic optimality of these sampling algorithms in terms of Smolyak sampling width \({r^{s}_{n}}({U^{r}_{p}}, L_{q})\) and sampling width \(r_{n}({U^{r}_{p}}, L_{q})\).

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References

  1. Besov, O.V.: Multiplicative estimates for integral norms of differentiable functions of several variables. Proc. Steklov Inst. Math. 131, 1–14 (1974)

    MathSciNet  MATH  Google Scholar 

  2. Besov, O.V., Il’in, V.P., Nikol’skii, S.M.: Integral Representations of Functions and Imbedding Theorems, vol. 1. Halsted Press, New York (1978)

    MATH  Google Scholar 

  3. Bokanowski, O., Garcke, J., Griebel, M., Klompmaker, I.: An adaptive sparse grid semi-Lagrangian scheme for first order Hamilton-Jacobi Bellman equations. J. Sci. Comput. 55(3), 575–605 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bungartz, H.-J., Griebel, M.: Sparse grids. Acta Numer. 13, 147–269 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Byrenheid, G., Dũng, D., Sickel, W., Ullrich, T.: Sampling on energy-norm based sparse grids for the optimal recovery of Sobolev type functions in H γ. J. Approx. Theory 207, 207–231 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Byrenheid, G., Ullrich, T.: Optimal sampling recovery of mixed order Sobolev embeddings via discrete Littlewood–Paley type characterizations. arXiv:1603.04809 (2016)

  7. Byrenheid, G., Ullrich, T.: The Faber-Schauder system in spaces with bounded mixed derivative and nonlinear approximation. Manuscript (2016)

  8. Chui, C.K.: An Introduction to Wavelets. Academic Press, New York (1992)

    MATH  Google Scholar 

  9. de Bore, C., Höllig, K., Riemenschneider, S.: Box Spline. Springer, Berlin (1993)

    Book  Google Scholar 

  10. Dũng, D.: On recovery and one-sided approximation of periodic functions of several variables. Dokl. Akad. SSSR 313, 787–790 (1990)

    Google Scholar 

  11. Dũng, D.: On optimal recovery of multivariate periodic functions. In: Igary, S. (ed.) Harmonic Analysis (Conference Proceedings), pp. 96–105. Springer, Tokyo-Berlin (1991)

  12. Dũng, D.: Optimal recovery of functions of a certain mixed smoothness. Vietnam J. Math. 20(2), 18–32 (1992)

    MathSciNet  MATH  Google Scholar 

  13. Dũng, D.: Continuous algorithms in n-term approximation and non-linear widths. J. Approx. Theory. 102, 217–242 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dũng, D.: Non-linear approximations using sets of finite cardinality or finite pseudo-dimension. J. Complex. 17, 467–492 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dũng, D.: Non-linear sampling recovery based on quasi-interpolant wavelet representations. Adv. Comput. Math. 30, 375–401 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dũng, D.: Optimal adaptive sampling recovery. Adv. Comput. Math. 34, 1–41 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dũng, D.: B-spline quasi-interpolant representations and sampling recovery of functions with mixed smoothness. J. Complex. 27, 541–467 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dũng, D.: Sampling and cubature on sparse grids based on a B-spline quasi-interpolation. Found. Comp. Math. 16, 1193–1240 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Dũng, D., Temlyakov, V.N., Ullrich, T.: Hyperbolic cross approximation. arXiv:1601.03978[math.NA] (2015)

  20. Dũng, D., Ullrich, T.: Lower bounds for the integration error for multivariate functions with mixed smoothness and optimal Fibonacci cubature for functions on the square. Math. Nachr. 288, 743–762 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Fefferman, C., Stein, E.M.: Some maximal inequalities. Am. J. Math. 93, 107–115 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  22. Galeev, E.M.: On linear widths of classes of periodic functions of several variables. Vestnik MGU Ser.1 Mat.-Mekh. 4, 13–16 (1987)

    MathSciNet  Google Scholar 

  23. Galeev, E.M.: Linear widths of Hölder-Nikol’skii classes of periodic functions of several variables. Mat. Zametki 59, 189–199 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  24. Garcke, J., Hegland, M.: Fitting multidimensional data using gradient penalties and the sparse grid combination technique. Computing 84(1-2), 1–25 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gerstner, T., Griebel, M.: Sparse grids. In: Cont, R. (ed.) Encyclopedia of Quantitative Finance. Wiley, New York (2010)

  26. Griebel, M., Harbrecht, H.: A note on the construction of L-fold sparse tensor product spaces. Constr. Approx. 38(2), 235–251 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Griebel, M., Holtz, M.: Dimension-wise integration of high-dimensional functions with applications to finance. J. Complex. 26, 455–489 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Griebel, M., Harbrecht, H.: On the construction of sparse tensor product spaces. Math. Comput. 82(282), 975–994 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Jawerth, B.: Some observations on Besov and Lizorkin-Triebel spaces. Math. Scand. 40(1), 94–104 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  30. Griebel, M., Knapek, S.: Optimized general sparse grid approximation spaces for operator equations. Math. Comp. 78(268), 2223–2257 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Nikol’skaya, N.: Approximation of periodic functions in the class \(SH^{r}_{p}*\) by Fourier sums. Sibirsk. Mat. Zh. 16, 761–780 (1975). English transl. in Siberian Math. J. 16, 1975

    MATH  Google Scholar 

  32. Sickel, W., Ullrich, T.: The Smolyak algorithm, sampling on sparse grids and function spaces of dominating mixed smoothness. East J. Approx. 13, 387–425 (2007)

    MathSciNet  Google Scholar 

  33. Sickel, W., Ullrich, T.: Spline interpolation on sparse grids. Appl. Anal. 90, 337–383 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Schmeisser, H.J., Triebel, H.: Topics in Fourier Analysis and Function Spaces. Wiley, New York (1987)

    MATH  Google Scholar 

  35. Smolyak, S.A.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk 148, 1042–1045 (1963)

    MATH  Google Scholar 

  36. Temlyakov, V.: Approximation recovery of periodic functions of several variables. Mat. Sb. 128, 256–268 (1985)

    MathSciNet  Google Scholar 

  37. Temlyakov, V.N.: Approximation of periodic functions of several variables by trigonometric polynomials, and widths of some classes of functions. Izv. AN SSSR 49, 986–1030 (1985). English Transl. in Math. Izv. 27, 1986

    Google Scholar 

  38. Temlyakov, V.: On approximate recovery of functions with bounded mixed derivative. J. Complex. 9, 41–59 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  39. Temlyakov, V.: Approximation of Periodic Functions. Nova Science Publishers, New York (1993)

    MATH  Google Scholar 

  40. Triebel, H.: Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration. European Math. Soc. Publishing House, Zürich (2010)

    Book  MATH  Google Scholar 

  41. Ullrich, T.: Smolyak’s algorithm, sampling on sparse grids and Sobolev spaces of dominating mixed smoothness. East J. Approx. 14, 1–38 (2008)

    MathSciNet  MATH  Google Scholar 

  42. Ullrich, T.: Function spaces with dominating mixed smoothness, characterization by differences. Technical report, Jenaer Schriften zur Math. und Inform. Math/inf/05/06 (2006)

  43. Zenger, C.: Sparse grids. In: Hackbusch, W. (ed.) Parallel Algorithms for Partial Differential Equations, vol. 31 of Notes on Numerical Fluid Mechanics, Vieweg, Braunschweig/Wiesbaden (1991)

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Acknowledgements

This work is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 102.01-2017.05. A part of this work was done when the author was working as a research professor at the Vietnam Institute for Advanced Study in Mathematics (VIASM). He would like to thank the VIASM for providing a fruitful research environment and working condition. The author would like to thank Glenn Byrenheid and Tino Ullrich for giving opportunity to read the manuscript [7]. He thanks Glenn Byrenheid, Vladimir Temlyakov, and Tino Ullrich for useful discussions.

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Dũng, D. B-Spline Quasi-Interpolation Sampling Representation and Sampling Recovery in Sobolev Spaces of Mixed Smoothness. Acta Math Vietnam 43, 83–110 (2018). https://doi.org/10.1007/s40306-017-0230-3

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  • DOI: https://doi.org/10.1007/s40306-017-0230-3

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