Skip to main content
Log in

Basic Relations Valid for the Bernstein Space \(B^{p}_{\sigma}\) and Their Extensions to Functions from Larger Spaces with Error Estimates in Terms of Their Distances from \(B^{p}_{\sigma}\)

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript


There are various basic relations (equations and inequalities) that hold in Bernstein spaces \(B_{\sigma}^{p}\) but are no longer valid in larger spaces. However, when a function f is in some sense close to a Bernstein space, one may expect that the corresponding relation is not violated drastically. It should hold with a remainder that involves the distance of f from \(B_{\sigma}^{p}\).

First we establish a hierarchy of spaces that generalize the Bernstein spaces and are suitable for our studies. It includes Hardy spaces, Sobolev spaces, Lipschitz classes and Fourier inversion classes. Next we introduce an appropriate metric for describing the distance of a function belonging to such a space from a Bernstein space. It will be used for estimating remainders and studying rates of convergence.

In the main part, we present the desired extensions. Our considerations include the classical sampling formula by Whittaker-Kotel’nikov-Shannon, the sampling formula of Valiron-Tschakaloff, the differentiation formula of Boas, the reproducing kernel formula, the general Parseval formula, Bernstein’s inequality for the derivative and Nikol’skiĭ’s inequality estimating the \(l^{p}(\mathbb{Z})\) norm in terms of the \(L^{p}(\mathbb{R})\) norm. All the remainders are represented in terms of the Fourier transform of f and estimated from above by the new metric. Finally we show that the remainders can be continued to spaces where a Fourier transform need not exist and can be estimated in terms of the regularity of f.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1

Similar content being viewed by others


  1. Unfortunately there is a misprint in [18]. In No. 8.350(2), the lower limit of integration has to be replaced by x.


  1. Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Washington (1964)

    MATH  Google Scholar 

  2. Bakan, A., Kaijser, S.: Hardy spaces for the strip. J. Math. Anal. Appl. 333(1), 347–364 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bardaro, C., Butzer, P.L., Stens, R.L., Vinti, G.: Approximation error of the Whittaker cardinal series in terms of an averaged modulus of smoothness covering discontinuous signals. J. Math. Anal. Appl. 316(1), 269–306 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Boas, R.P.: The derivative of a trigonometric integral. J. Lond. Math. Soc. 12, 164–165 (1937)

    Article  MathSciNet  Google Scholar 

  5. Boas, R.P.: Entire Functions. Academic Press, New York (1954)

    MATH  Google Scholar 

  6. Brown, J.L. Jr.: On the error in reconstructing a non-bandlimited function by means of the bandpass sampling theorem. J. Math. Anal. Appl. 18, 75–84 (1967). Erratum ibid. 21, 699 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  7. Butzer, P.L., Ferreira, P.J.S.G., Higgins, J.R., Schmeisser, G., Stens, R.L.: The sampling theorem, Poisson’s summation formula, general Parseval formula, reproducing kernel formula and the Paley–Wiener theorem for bandlimited signals—their interconnections. Appl. Anal. 90, 431–461 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Butzer, P.L., Dodson, M., Ferreira, P.J.S.G., Higgins, J.R., Schmeisser, G., Stens, R.L.: The generalized Parseval decomposition formula, the approximate sampling theorem, the approximate reproducing kernel formula, Poisson’s summation formula and Riemann’s zeta function—their interconnections for non-bandlimited functions (2012). Manuscript

  9. Butzer, P.L., Gessinger, A.: The approximate sampling theorem, Poisson’s sum formula, a decomposition theorem for Parseval’s equation and their interconnections. Ann. Numer. Math. 4, 143–160 (1997)

    MathSciNet  MATH  Google Scholar 

  10. Butzer, P.L., Higgins, J.R., Stens, R.L.: Classical and approximate sampling theorems: studies in the \(L^{p}(\mathbb{R})\) and the uniform norm. J. Approx. Theory 137, 250–263 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Butzer, P.L., Nessel, R.J.: Fourier Analysis and Approximation, vol. 1. Birkhäuser, Basel (1971)

    Book  MATH  Google Scholar 

  12. Butzer, P.L., Nessel, R.J.: Work of De La Vallée Poussin in approximation theory and its impact. In: Butzer, P.L., Mawhin, J., Vetro, P. (eds.) De La Vallée Poussin, Charles-Jean: Collected Works/Œuvres Scientifiques, vol. III, pp. 375–414. Académie Royale de Belgique/Circolo Matematico di Palermo, Brussels/Palermo (2004)

    Google Scholar 

  13. Butzer, P.L., Scherer, K.: On the fundamental approximation theorems of D. Jackson, S.N. Bernstein, and theorems of M. Zamansky and S.B. Stečkin. Aequ. Math. 3, 170–185 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  14. Butzer, P.L., Scherer, K.: Über die Fundamentalsätze der klassischen Approximationstheorie in abstrakten Räumen. In: Butzer, P.L., Sz.-Nagy, B. (eds.) Abstract Spaces and Approximation, Proc. Conf. Math. Research Center, Oberwolfach, 1968. ISNM, vol. 10, pp. 113–125. Birkhäuser, Basel (1969)

    Google Scholar 

  15. Butzer, P.L., Splettstößer, W., Stens, R.L.: The sampling theorems and linear prediction in signal analysis. Jahresber. Dtsch. Math.-Ver. 90, 1–70 (1988)

    MATH  Google Scholar 

  16. DeVore, R.A., Lorentz, G.G.: Constructive Approximation. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  17. Gervais, R., Rahman, Q.I., Schmeisser, G.: Simultaneous interpolation and approximation. In: Sahney, B.N. (ed.) Polynomial and Spline Approximation, pp. 203–223. Reidel, Dordrecht (1979)

    Google Scholar 

  18. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products, 5th edn. Academic Press, Boston (1994)

    MATH  Google Scholar 

  19. Hayman, W.K., Kennedy, P.B.: Subharmanic Functions, vol. 1. Academic Press, London (1976)

    Google Scholar 

  20. Higgins, J.R.: Five short stories about the cardinal series. Bull. Am. Math. Soc. 12, 45–89 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  21. Higgins, J.R.: Sampling Theory in Fourier and Signal Analysis. Clarendon Press, Oxford (1996)

    MATH  Google Scholar 

  22. Junggeburth, J., Scherer, K., Trebels, W.: Zur besten Approximation auf Banachräumen mit Anwendungen auf ganze Funktionen. Forschungsber. Landes Nordrh.-Westfal. 2311, 51–75 (1973)

    MathSciNet  Google Scholar 

  23. Kincaid, D., Cheney, W.: Numerical Analysis, 2nd edn. Brooks/Cole Publishing Company, Pacific Grove (1996)

    MATH  Google Scholar 

  24. Nikol’skiĭ, S.M.: Approximation of Functions of Several Variables and Imbedding Theorems. Springer, Berlin (1975)

    Book  Google Scholar 

  25. Rahman, Q.I., Schmeisser, G.: On a Gaussian quadrature formula for entire functions of exponential type. In: Collatz, L., Meinardus, G., Nürnberger, G. (eds.) Numerical Methods in Approximation Theory VIII, September–October, 1986. ISNM, vol. 81, pp. 169–183. Birkhäuser, Basel (1987)

    Google Scholar 

  26. Schmeisser, G.: Numerical differentiation inspired by a formula of R.P. Boas. J. Approx. Theory 160, 202–222 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Stenger, F.: Numerical methods based on Whittaker cardinal, or sinc functions. SIAM Rev. 23, 165–224 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  28. Stenger, F.: Numerical Methods Based on Sinc and Analytic Functions. Springer, New York (1993)

    Book  MATH  Google Scholar 

  29. Titchmarsh, E.C.: A note on Fourier transforms. J. Lond. Math. Soc. 2, 148–150 (1927)

    Article  MathSciNet  MATH  Google Scholar 

  30. Titchmarsh, E.C.: The Theory of Functions, 2nd edn. Oxford University Press, Oxford (1939)

    MATH  Google Scholar 

  31. Titchmarsh, E.C.: Introduction to the Theory of Fourier Integrals, 2nd edn. Clarendon Press, Oxford (1948)

    Google Scholar 

  32. Tschakaloff, L.: Zweite Lösung der Aufgabe 105. Jahresber. Dtsch. Math.-Ver. 43, 11–12 (1933)

    Google Scholar 

  33. Valiron, G.: Sur la formule d’interpolation de Lagrange. Bull. Sci. Math. 49, 181–192, 203–224 (1925)

    Google Scholar 

  34. Weiss, P.: An estimate of the error arising from misapplication of the sampling theorem. Not. Am. Math. Soc. 10, 351 (1963)

    Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Gerhard Schmeisser.

Additional information

Communicated by Hans G. Feichtinger.

A preliminary version of this paper was presented by one of the authors (G. Schmeisser) as an invited one hour lecture at the First Bavaria-Québec Mathematical Meeting held at Montreal (November 30 to December 3, 2009) and conducted by R. Fournier and St. Ruscheweyh. The paper in its present form, but without the new results on derivative-free error estimates, was the basis of an invited lecture given jointly by two of the authors (P.L. Butzer and G. Schmeisser) at the workshop From Abstract to Computational Harmonic Analysis, held at Strobl (Austria, June 13–19, 2011) in honour of the 60th birthday of H.G. Feichtinger; this workshop was conducted by K. Gröchenig and T. Strohmer.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Butzer, P.L., Schmeisser, G. & Stens, R.L. Basic Relations Valid for the Bernstein Space \(B^{p}_{\sigma}\) and Their Extensions to Functions from Larger Spaces with Error Estimates in Terms of Their Distances from \(B^{p}_{\sigma}\) . J Fourier Anal Appl 19, 333–375 (2013).

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification