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Multivariate rational approximation


Authors: Ronald A. DeVore and Xiang Ming Yu
Journal: Trans. Amer. Math. Soc. 293 (1986), 161-169
MSC: Primary 41A20; Secondary 41A63
DOI: https://doi.org/10.1090/S0002-9947-1986-0814918-6
MathSciNet review: 814918
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Abstract: We estimate the error in approximating a function $ f$ by rational functions of degree $ n$ in the norm of $ {L_q}(\Omega ),\,\Omega : = {[0,\,1]^d}$. Among other things, we prove that if $ f$ is in the Sobolev space $ W_p^k(\Omega )$ and if $ k/d - 1/p + 1/q > 0$, then $ f$ can be approximated by rational functions of degree $ n$ to an order $ O({n^{ - k/d}})$.


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  • [1] Yu. Brudnyi, Rational approximation and exotic Lipschitz spaces, in [4, pp. 25-30]. MR 588168 (82f:41022)
  • [2] M. S. Birman and M. Solomjak, Piecewise-polynomial approximation of functions of the classes $ W_p^\alpha $, Math. USSR-Sb. 2 (1967), 295-317. MR 0217487 (36:576)
  • [3] R. A. DeVore, Maximal functions and their application to rational approximation, CBMS Regional Conf. Ser. Math., no. 3, Amer. Math. Soc., Providence, R. I., 1983, pp. 143-155. MR 729327 (85g:41022)
  • [4] R. A. DeVore and K. Scherer (editors), Quantitative approximation, Academic Press, New York, 1980. MR 588164 (81i:41001)
  • [5] R. A. DeVore and R. C. Sharpley, Maximal functions measuring smoothness, Mem. Amer. Math. Soc., Vol. 47, No. 293, 1984. MR 727820 (85g:46039)
  • [6] G. G. Lorentz, Approximation of functions, Holt, New York, 1966, p. 94. MR 0213785 (35:4642)
  • [7] D. J. Newman, Rational approximation to $ \vert x\vert$, Michigan Math. J. 11 (1964), 11-14. MR 0171113 (30:1344)
  • [8] V. Popov, Uniform rational approximation of the class $ {V_r}$ and its applications, Acta Math. Acad. Sci. Hungar. 29 (1977), 119-129. MR 0435679 (55:8637)
  • [9] -, On the connection between rational uniform approximation and polynomial $ {L_p}$ approximation of functions, in [4, 267-277]. MR 588187 (81j:41025)
  • [10] S. L. Sobolev, Applications of functional analysis in mathematical physics, Transl. Math. Monos., vol. 7, Amer. Math. Soc., Providence, R. I., 1963.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1986-0814918-6
Keywords: Multivariate rational approximation, error estimates in $ {L_q}$, Sobolev spaces
Article copyright: © Copyright 1986 American Mathematical Society

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