Approximation of Non-decaying Signals from Shift-Invariant Subspaces

  • Ha Q. NguyenEmail author
  • Michael Unser


In our recent work, the sampling and reconstruction of non-decaying signals, modeled as members of weighted-\(L_p\) spaces, were shown to be stable with an appropriate choice of the generating kernel for the shift-invariant reconstruction space. In this paper, we extend the Strang–Fix theory to show that, for d-dimensional signals whose derivatives up to order L are all in some weighted-\(L_p\) space, the weighted norm of the approximation error can be made to go down as \(O(h^L)\) when the sampling step h tends to 0. The sufficient condition for this decay rate is that the generating kernel belongs to a particular hybrid-norm space and satisfies the Strang–Fix conditions of order L. We show that the \(O(h^L)\) behavior of the error is attainable for both approximation schemes using projection (when the signal is prefiltered with the dual kernel) and interpolation (when a prefilter is unavailable). The requirement on the signal for the interpolation method, however, is slightly more stringent than that of the projection because we need to increase the smoothness of the signal by a margin of \(d/p+\varepsilon \), for arbitrary \(\varepsilon >0\). This extra amount of derivatives is used to make sure that the direct sampling is stable.


Approximation theory Strang–Fix conditions Shift-invariant spaces Spline interpolation Weighted \(L_p\) Weighted Sobolev spaces Hybrid-norm spaces 


  1. 1.
    Aimar, H.A., Bernardis, A.L., Martín-Reyes, F.J.: Multiresolution approximations and wavelet bases of weighted \(L^p\) spaces. J. Fourier Anal. Appl. 9(5), 497–510 (2003)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Bardaro, C., Butzer, P., Stens, R., 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)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Blu, T., Unser, M.: Approximation error for quasi-interpolators and (multi-) wavelet expansions. Appl. Comput. Harmon. Anal. 6(2), 219–251 (1999)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Blu, T., Unser, M.: Quantitative Fourier analysis of approximation: Part I. Interpolators and projectors. IEEE Trans. Signal Process. 47(10), 2783–2795 (1999)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Blu, T., Unser, M.: Quantitative Fourier analysis of approximation: Part II–Wavelets. IEEE Trans. Signal Process. 47(10), 2796–2806 (1999)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Chui, C.K.: Multivariate Splines. Society of Industrial and Applied Mathematics, Philadelphia, PA (1988)zbMATHGoogle Scholar
  7. 7.
    Chui, C.K., Diamond, H.: A characterization of multivariate quasi-interpolation formulas and applications. Numer. Math. 57, 105–121 (1990)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Davis, P.J.: Interpolation and Approximation. Dover, New York, NY (1975)zbMATHGoogle Scholar
  9. 9.
    de Boor, C.: A Practical Guide to Splines. Springer-Verlag, New York, NY (1978)zbMATHGoogle Scholar
  10. 10.
    de Boor, C.: The polynomials in the linear span of integer translates of a compactly supported function. Constr. Approx. 3, 199–208 (1987)MathSciNetzbMATHGoogle Scholar
  11. 11.
    de Boor, C.: Quasi-interpolants and approximation power of multivariate splines. In: Dahmen, W., Gasca, M., Micchelli, C.A. (eds.) Computation of Curves and Surfaces, pp. 313–345. Kluwer, Dordrecht (1990)Google Scholar
  12. 12.
    de Boor, C., Fix, G.: Spline approximation by quasi-interpolants. J. Approx. Theory 8, 19–45 (1973)zbMATHGoogle Scholar
  13. 13.
    de Boor, C., Jia, R.-Q.: Controlled approximation and a characterization of the local approximation order. Proc. Amer. Math. Soc. 95(4), 547–553 (1985)MathSciNetzbMATHGoogle Scholar
  14. 14.
    de Boor, C., DeVore, R.A., Ron, A.: Approximation from shift-invariant subspaces of \(L_2(\mathbb{R}^d)\). Trans. Amer. Math. Soc. 341(2), 787–806 (1994)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Dragotti, P., Vetterli, M., Blu, T.: Sampling moments and reconstructing signals of finite rate of innovation: Shannon meets Strang-Fix. IEEE Trans. Signal Process. 55(5), 1741–1757 (2007)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Edmunds, D.E., Kokilashvili, V., Meskhi, A.: On Fourier multipliers in weighted Triebel-Lizorkin spaces. J. Inequal. Appl. 74(4), 555–591 (2002)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Fefferman, C., Stein, E.M.: Some maximal inequalities. Amer. J. Math. 93(1), 107–115 (1971)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Feichtinger, H.G.: New results on regular and irregular sampling based on Wiener amalgams. In: Jarosz, K. (ed.) Proceedings of the International Conference on Function Spaces, series: Lecture Notes in Pure and Applied Mathematics, pp. 107–121. Dekker, New York (1991)Google Scholar
  19. 19.
    Gel’fand, I., Raikov, D., Shilov, G.: Commutative Normed Rings. Chelsea Publishing Co., Hartford (1964)zbMATHGoogle Scholar
  20. 20.
    Grafakos, L.: Modern Fourier Analysis, 2nd edn. Springer, New York (2008)zbMATHGoogle Scholar
  21. 21.
    Grafakos, L.: Classical Fourier Analysis, 2nd edn. Springer, New York (2008)zbMATHGoogle Scholar
  22. 22.
    Gröchenig, K.: Weight functions in time-frequency analysis. arXiv:math/0611174 [math.FA], (2006)
  23. 23.
    Hardy, G.H., Littlewood, J.E.: A maximal theorem with function-theoretic applications. Acta Math. 54(1), 81–116 (1930)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Heil, C.: An introduction to weighted Wiener amalgams. In: Krishna, M., Radha, R., Thangavelu, S. (eds.) Wavelets and Their Applications, pp. 183–216. Allied Publishers, New Delhi (2003)Google Scholar
  25. 25.
    Jia, R.Q.: Approximation by quasi-projection operators in Besov spaces. J. Approx. Theory 162(1), 186–200 (2010)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Jia, R.-Q., Lei, J.: Approximation by multiinteger translates of functions having global support. J. Approx. Theory 72(1), 2–23 (1993)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Kolomoitsev, Y., Krivoshein, A., Skopina, M.: Differential and falsified sampling expansions. J. Fourier Anal. Appl. (2017).
  28. 28.
    Kolomoitsev, Y., Skopina, M.: Approximation by multivariate Kantorovich–Kotelnikov operators. J. Math. Anal. Appl. 456(1), 195–213 (2017)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Krivoshein, A., Skopina, M.: Multivariate sampling-type approximation. Anal. Appl. 15(4), 521–542 (2017)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Król, S.: Fourier multipliers on weighted \(L^p\) spaces, arXiv:1403.4477 [math.CA], (2014)
  31. 31.
    Kurtz, D.S.: Littlewood-Paley and multiplier theorems on weighted \(L^p\) spaces. Trans. Amer. Math. Soc. 259(1), 235–254 (1980)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Lei, J.: \(L_p\)-approximation by certain projection operators. J. Math. Anal. Appl. 185(1), 1–14 (1994)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Light, W.A.: Recent developments in the Strang-Fix theory for approximation orders. In: Laurent, P.J., Méhauté, A.L., Schumaker, L.L. (eds.) Curves and Surfaces, pp. 285–292. Academic Press, Boston, MA (1991)Google Scholar
  34. 34.
    Light, W.A., Cheney, E.W.: Quasi-interpolation with translates of a function having noncompact support. Constr. Approx 8(1), 35–48 (1992)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165, 207–226 (1972)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Nguyen, H.Q., Unser, M.: Generalized Poisson summation formula for tempered distributions, In: Proceedings of the 11th International Conference on Sampling Theory and Applications (SampTA’15), pp. 1–5 25–29 May 2015Google Scholar
  37. 37.
    Nguyen, H.Q., Unser, M.: A sampling theory for non-decaying signals. Appl. Comput. Harmon. Anal. 43(1), 76–93 (2017)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Nguyen, H.Q., Unser, M., Ward, J.-P.: Generalized Poisson summation formulas for continuous functions of polynomial growth. J. Fourier Anal. Appl. 23(2), 442–461 (2017)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Schoenberg, I.J.: Contributions to the problem of approximation of equidistant data by analytic functions. Quart. Appl. Math. 4(45–99), 112–141 (1946)MathSciNetGoogle Scholar
  40. 40.
    Schoenberg, I.J.: Cardinal Spline Interpolation. Society of Industrial and Applied Mathematics, Philadelphia, PA (1973)zbMATHGoogle Scholar
  41. 41.
    Schumaker, L.L.: Spline Functions: Basic Theory. Wiley, New York, NY (1981)zbMATHGoogle Scholar
  42. 42.
    Shannon, C.E.: Communication in the presence of noise. Proc. IRE 37(1), 10–21 (1949)MathSciNetGoogle Scholar
  43. 43.
    Stein, E.M.: On certain operators on \(L_p\) spaces, Ph.D. Dissertation, University of Chicago, Chicago, IL, (1955)Google Scholar
  44. 44.
    Strang, G., Fix, G.: A Fourier analysis of the finite element variational method. In: Geymonat, G. (ed.) Constructive Aspects of Functional Analysis, pp. 796–830. Springer, Rome (1971)Google Scholar
  45. 45.
    Tomita, N.: Strang-Fix theory for approximation order in weighted \(L^p\)-spaces and Herz spaces. J. Funct. Space Appl. 4(1), 7–24 (2006)zbMATHMathSciNetGoogle Scholar
  46. 46.
    Unser, M.: Quasi-orthogonality and quasi-projections. Appl. Comput. Harmon. Anal. 3(3), 201214 (1996)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Unser, M.: Sampling–50 years after Shannon. Proc. IEEE 88(4), 569–587 (2000)zbMATHGoogle Scholar
  48. 48.
    Unser, M., Daubechies, I.: On the approximation power of convolution-based least squares versus interpolation. IEEE Trans. Signal Process. 45(7), 1697–1711 (1997)zbMATHGoogle Scholar
  49. 49.
    Unser, M., Tafti, P.D.: An Introduction to Sparse Stochastic Processes. Cambridge University Press, Cambridge (2014)zbMATHGoogle Scholar
  50. 50.
    Unser, M., Aldroubi, A., Eden, M.: B-spline signal processing: Part I–theory. IEEE Trans. Signal Process. 41(2), 821–833 (1993)zbMATHGoogle Scholar
  51. 51.
    Unser, M., Aldroubi, A., Eden, M.: B-spline signal processing: Part II—efficiency design and applications. IEEE Trans. Signal Process. 41(2), 834–848 (1993)zbMATHGoogle Scholar
  52. 52.
    Unser, M., Tafti, P.D., Sun, Q.: A unified formulation of Gaussian versus sparse stochastic processes–Part I: continuous-domain theory. IEEE Trans. Inform. Theory 60(3), 1945–1962 (2014)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Unser, M., Tafti, P.D., Amini, A., Kirshner, H.: A unified formulation of Gaussian versus sparse stochastic processes–Part II: discrete-domain theory. IEEE Trans. Inform. Theory 60(5), 3036–3051 (2014)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Wiener, N.: The Fourier Integral and Certain of its Applications. MIT Press, Cambridge (1933)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Viettel Research and Development InstituteHanoiVietnam
  2. 2.Biomedical Imaging GroupÉcole Polytechnique Fédérale de Lausanne (EPFL)LausanneSwitzerland

Personalised recommendations