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Riesz basis, Paley-Wiener class and tempered splines

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Abstract

The Marcinkiewicz-Zygmund inequality and the Bernstein inequality are established on ℒ2m (T, ℝ) ∩ L2 (ℝ) which is the space of polynomial splines with irregularly distributed nodesT = {t j } j ∈ℤ, where {t j }j∈ℤ is a real sequence such that {eitξ} j }j ∈ℤ constitutes a Riesz basis for L2([ −π,π]). From these results, the asymptotic relation

is proved, where B π,2 denotes the set of all functions from L2( R) which can be continued to entire functions of exponential type ⪯ ϕ, i.e. the classical Paley-Wiener class.

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References

  1. Akhieser, N. I., Theory of Approximation, New York:Ungar, 1956.

    Google Scholar 

  2. Lyubarskii, Y., Madych, W. R., The recovery of irregular sampled band limited functions via tempered splines, J. of Functional Analysis, 1994, 125:201.

    Article  MATH  MathSciNet  Google Scholar 

  3. Seip, K., On the connection between exponential bases and certain related sequence in L2 - ϕ, ϕ), J. of Functional Analysis, 1995, 130:131.

    Article  MATH  MathSciNet  Google Scholar 

  4. DeVore, R. A., Lorentz, G. G., Constructive Approximation, Berlin:Springer-Verlag, 1993.

    MATH  Google Scholar 

  5. Yong, R., An Introduction to Nonhannonic Fourier Series, New York:Academic Press, 1980.

    Google Scholar 

  6. Zygmund, A., Trigonometric Series, 2nd ed., London:Cambridge Univ. Press, 1959.

    MATH  Google Scholar 

  7. Richards, F. B., Schoenberg, J. J., Notes on spline functions IV: A cardinal spline analogue of the theorem of brothers Markov, Israel J. Math., 1973, 16:94.

    Article  MATH  MathSciNet  Google Scholar 

  8. Fang Gensun, Approximation properties of entire functions of exponential type, J. Math. Anal. Appl., 1996, 201:642.

    Article  MATH  MathSciNet  Google Scholar 

  9. Fang Gensun, On an asymptotic connection between approximation by the cardinal splines and entire functions of exponential type, Chinese Science Bulletin, 1996, 41(4):265.

    MATH  MathSciNet  Google Scholar 

  10. Korneichuk, N., Exact Constants in Approximation Theory, Cambridge:Cambridge Univ. Press, 1991.

    MATH  Google Scholar 

  11. Timan, A. F., Theory of Approximation of Functions of a Real Variable, New York:MacMillan, 1963.

    MATH  Google Scholar 

  12. Tikhomirov, V. M., Some Question of Approximation Theory, Moscow:Izdat Moscow 6s. Univ., 1976.

    Google Scholar 

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Correspondence to Gensun Fang.

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Fang, G., Long, J. Riesz basis, Paley-Wiener class and tempered splines. Sci. China Ser. A-Math. 43, 1075–1082 (2000). https://doi.org/10.1007/BF02898242

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  • DOI: https://doi.org/10.1007/BF02898242

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