Acta Mathematica Hungarica

, Volume 128, Issue 1–2, pp 26–35 | Cite as

Approximation of bandlimited functions by trigonometric polynomials

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Abstract

Let σ > 0. For 1 ≦ p ≦ ∞, the Bernstein space B σ p is a Banach space of all f ∈ L p (ℝ) such that f is bandlimited to σ; that is, the distributional Fourier transform of f is supported in [−σ,σ]. We study the approximation of f ∈ B σ p by finite trigonometric sums
$$ P_\tau (x) = \chi _\tau (x) \cdot \sum\limits_{|k| \leqq \sigma \tau /\pi } {c_{k,\tau } e^{i\frac{\pi } {\tau }kx} } $$
in L p norm on ℝ as τ → ∞, where χ τ denotes the indicator function of [−τ, τ].

Key words and phrases

Fourier transform bandlimited function entire function of exponential type exponential sum trigonometric polynomial Bernstein space approximation by trigonometric polynomial 

2000 Mathematics Subject Classification

primary 30D15 secondary 42A10 
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References

  1. [1]
    N. Y. Achieser, Theory of Approximation, Dover Publications. Inc. (New York, 1992).Google Scholar
  2. [2]
    E. Hewitt and K. A. Ross, Abstract Harmonic Analysis II, Springer-Verlag (Berlin - Heidelberg, 1997).Google Scholar
  3. [3]
    J. R. Higgins, Sampling Theory in Fourier and Signal Analysis: Foundations, Clarendon Press (Oxford, 1996).MATHGoogle Scholar
  4. [4]
    L. Hörmander, A new proof and a generalization of an inequality of Bohr, Math. Scand., 2 (1954), 33–45.MATHMathSciNetGoogle Scholar
  5. [5]
    L. Hörmander, Some inequalities for functions of exponential type, Math. Scand., 3 (1955), 21–27.MATHMathSciNetGoogle Scholar
  6. [6]
    M. G. Krein, On the representation of functions by Fourier-Stieltjes integrals (in Russian), Uch. Zap. Kuibysh. Gos. Ped. Uchit. Inst., 7 (1943), 123–147.Google Scholar
  7. [7]
    B. Ya. Levin, Lectures on Entire Functions, Transl. Math. Monogr., 150, Amer. Math. Soc. (1996).Google Scholar
  8. [8]
    B. M. Lewitan, Über eine Verallgemeinerung der Ungleichungen von S. Bernstein und H. Bohr, Dokl. Akad. Nauk, 15 (1937), 169–172.Google Scholar
  9. [9]
    R. Martin, Approximation of Ω-bandlimited functions by Ω-bandlimited trigonometric polynomials, Sampl. Theory Signal Image Process., 6 (2007), 273–296.MATHMathSciNetGoogle Scholar
  10. [10]
    G. V. Milovanović, D. S. Mitrinović and Th. M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific Publishing Co., Inc. (River Edge, NJ, 1994).MATHGoogle Scholar
  11. [11]
    S. Norvidas, Approximation of entire functions by exponential polynomials, Lith. Math. J., 34 (1994), 415–421.CrossRefMathSciNetGoogle Scholar
  12. [12]
    S. Norvidas, Approximation of bandlimited functions by finite exponential sums, Lith. Math. J., 49 (2009), 185–189.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Q. I. Rahman and G. Schmeisser, L p inequalities for entire functions of exponential type, Trans. Amer. Math. Soc., 320 (1990), 91–103.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    G. Schmeisser, Approximation of entire functions of exponential type by trigonometric polynomials, Sampl. Theory Signal Image Process., 6 (2007), 297–306.MATHMathSciNetGoogle Scholar
  15. [15]
    A. F. Timan, Theory of Approximation of Functions of a Real Variable, Dover Publications, Inc. (New York, 1994).Google Scholar
  16. [16]
    R. M. Trigub and E. S. Belinsky, Fourier Analysis and Approximation of Functions, Kluwer Academic Publishers (Dordrecht, 2004).MATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2010

Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsVilniusLithuania

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