Skip to main content
SpringerLink
Log in

Spline wavelet bases in function spaces with Muckenhoupt weights

  • Published:
Revista Matemática Complutense Aims and scope Submit manuscript

Abstract

Properties of a class of spline wavelet systems of Battle–Lemarié type are described and applied to wavelet characterisation of Besov spaces and Triebel–Lizorkin spaces with Muckenhoupt weights.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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, 497–510 (2003)

    MathSciNet  MATH  Google Scholar 

  2. Battle, G.: A block spin construction of ondelettes, Part I: Lemarié functions. Commun. Math. Phys. 110, 601–615 (1987)

    Google Scholar 

  3. Battle, G.: A block spin construction of ondelettes, Part II: QFT connection. Commun. Math. Phys. 114, 93–102 (1988)

    MathSciNet  Google Scholar 

  4. Berkolaiko, M.Z., Novikov, I.Y.: Unconditional bases in spaces of functions of anisotropic smoothness. Proc. Steklov Inst. Math. 204, 27–41 (1994)

    MathSciNet  Google Scholar 

  5. Besov, O.V., Il’in, V.P., Nikol’skii, S.M.: Integral Representations of Functions and Imbedding Theorems, vol. I and II. V.H. Winston & Sons, Washington (1979)

    MATH  Google Scholar 

  6. Bourdaud, G.: Ondelettes et espaces de Besov. Rev. Mat. Iberoamericana 1(1), 477–512 (1995)

    MathSciNet  MATH  Google Scholar 

  7. Bownik, M.: Atomic and molecular decompositions of anisotropic Besov spaces. Math. Z. 250, 539–571 (2005)

    MathSciNet  MATH  Google Scholar 

  8. Bownik, M., Ho, K.-P.: Atomic and molecular decompositions of anisotropic Triebel–Lizorkin spaces. Trans. Amer. Math. Soc. 358(4), 1469–1510 (2005)

    MathSciNet  MATH  Google Scholar 

  9. Bui, H.-Q.: Weighted Besov and Triebel–Lizorkin spaces: interpolation by the real method. Hiroshima Math. J. 12, 581–605 (1982)

    MathSciNet  MATH  Google Scholar 

  10. Chui, C.K.: An Introduction to Wavelets. Academic Press, New York (1992)

    MATH  Google Scholar 

  11. Ciesielski, Z.: Constructive function theory and spline systems. Studia Math. 53(3), 277–302 (1975)

    MathSciNet  MATH  Google Scholar 

  12. Ciesielski, Z.: Equivalence, unconditionality and convergence a.e. of the spline bases in \(L_p\) spaces. In: Approximation Theory. Vol. 4. Banach Center Publications, pp. 55–68 (1979)

  13. Ciesielski, Z., Figiel, T.: Spline approximation and Besov spaces on compact manifolds. Studia Math. 75(1), 13–36 (1982)

    MathSciNet  MATH  Google Scholar 

  14. Ciesielski, Z., Figiel, T.: Spline bases in classical function spaces on compact \(C^{\infty }\) manifolds, Part I. Studia Math. 76(1), 1–58 (1983)

    MathSciNet  MATH  Google Scholar 

  15. Ciesielski, Z., Figiel, T.: Spline bases in classical function spaces on compact \(C^{\infty }\) manifolds, Part II. Studia Math. 76(2), 95–136 (1983)

    MathSciNet  MATH  Google Scholar 

  16. Daubechies, I.: Ten lectures on wavelets. SIAM (1992)

  17. Edmunds, D.E., Triebel, H.: Function Spaces, Entropy Numbers, Differential Operators. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  18. Farkas, W.: Atomic and subatomic decompositions in anisotropic function spaces. Math. Nachr. 209, 83–113 (2000)

    MathSciNet  MATH  Google Scholar 

  19. Farwig, R., Sohr, H.: Weighted \(L^q-\) theory for the Stokes resolvent in exterior domains. J. Math. Soc. Jpn. 49(2), 251–288 (1997)

    MATH  Google Scholar 

  20. Frazier, M., Jawerth, B.: Decomposition of Besov spaces. Indiana Univ. Math. J. 34(4), 777–799 (1985)

    MathSciNet  MATH  Google Scholar 

  21. Frazier, M., Jawerth, B.: A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93, 34–170 (1990)

    MathSciNet  MATH  Google Scholar 

  22. Frazier, M., Jawerth, B., Weiss, G.: Littlewood–Paley theory and the study of function spaces. Memoirs of the American Mathematical Society, vol. 79, Amer. Math. Soc., Providence, RI (1991)

  23. García-Cuerva, J., Kazarian, K.S.: Calderón-Zygmund operators and unconditional bases of weighted Hardy spaces. Studia Math. 109(3), 255–276 (1994)

    MathSciNet  MATH  Google Scholar 

  24. García-Cuerva, J., Kazarian, K.S.: Spline wavelet bases of weighted \(L^p\) spaces, \(1\le p <\infty \). Proc. Amer. Math. Soc. 123(2), 433–439 (1995)

    MathSciNet  MATH  Google Scholar 

  25. Haroske, D.D., Piotrowska, I.: Atomic decompositions of function spaces with Muckenhoupt weights, and some relation to fractal analysis. Math. Nachr. 281(10), 1476–1494 (2008)

    MathSciNet  MATH  Google Scholar 

  26. Haroske, D.D., Skandera, Ph, Triebel, H.: An approach to wavelet isomorphisms of function spaces via atomic representations. J. Fourier Anal. Appl. 24(3), 830–871 (2018)

    MathSciNet  MATH  Google Scholar 

  27. Haroske, D., Skrzypczak, L.: Entropy and approximation numbers of embeddings in function spaces with Muckenhoupt weights. I. Rev. Mat. Complut. 21(1), 135–177 (2008)

    MathSciNet  MATH  Google Scholar 

  28. Haroske, D., Triebel, H.: Entropy numbers in weighted function spaces and eigenvalue distributions of some degenerate pseudodifferential operators, I. Math. Nachr. 167, 131–156 (1994)

    MathSciNet  MATH  Google Scholar 

  29. Haroske, D., Triebel, H.: Entropy numbers in weighted function spaces and eigenvalue distributions of some degenerate pseudodifferential operators, II. Math. Nachr. 168, 109–137 (1994)

    MathSciNet  MATH  Google Scholar 

  30. Haroske, D., Triebel, H.: Wavelet bases and entropy numbers in weighted function spaces. Math. Nachr. 278, 108–132 (2005)

    MathSciNet  MATH  Google Scholar 

  31. Hörmander, L.: The Analysis of Linear Partial Differential Operators, I. Distribution Theory and Fourier Analysis. Springer, Berlin (1990)

    MATH  Google Scholar 

  32. Izuki, M., Sawano, Y.: Wavelet basis in the weighted Besov and Triebel–Lizorkin spaces with \(A_p^{{\rm loc}}\) weights. J. Approx. Theory 161, 656–673 (2009)

    MathSciNet  MATH  Google Scholar 

  33. Izuki, M., Sawano, Y.: Atomic decomposition for weighted Besov/Triebel–Lizorkin spaces with local class of weights. Math. Nachr. 285, 103–126 (2012)

    MathSciNet  MATH  Google Scholar 

  34. Lemarié, P.G.: Une nouvelle base d’ondelettes de \(L^2({\mathbb{R}}^N)\). J. de Math. Pures et Appl. 67, 227–236 (1988)

    MathSciNet  Google Scholar 

  35. Lemarié-Rieusset, P.G.: Ondelettes et poids de Muckenhoupt. Studia Math. 108(2), 127–147 (1994)

    MathSciNet  MATH  Google Scholar 

  36. Mallat, S.: A Wavelet Tour of Signal Processing, 2nd edn. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  37. Malecka, A.: Haar functions in weighted Besov and Triebel–Lizorkin spaces. J. Approx. Theory 200, 1–27 (2015)

    MathSciNet  MATH  Google Scholar 

  38. Meyer, Y.: Wavelets and Operators. Cambridge University Press, Cambridge (1992)

    MATH  Google Scholar 

  39. Muckenhoupt, B.: Hardy’s inequality with weights. Studia Math. 44, 31–38 (1972)

    MathSciNet  MATH  Google Scholar 

  40. Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165, 207–226 (1972)

    MathSciNet  MATH  Google Scholar 

  41. Muckenhoupt, B.: The equivalence of two conditions for the weight functions. Studia Math. 49, 101–106 (1973/74)

  42. Nasyrova, M.G., Ushakova, E.P.: Wavelet basis and entropy numbers of Hardy operator. Anal. Math. 44(4), 543–576 (2018)

    MathSciNet  MATH  Google Scholar 

  43. Novikov, I.Ya., Protasov, V.Yu., Skopina, M.A.: Wavelet theory, AMS, Translations Mathematical Monographs, vol. 239, (2011). (In Russian: I.Ya. Novikov, V.Yu. Protasov, M.A. Skopina, Teoriya vspleskov, FIZMATLIT, 2005)

  44. Novikov, IYa., Stechkin, S.B.: Basic constructions of wavelets. Fundam. Prikl. Mat. 3(4), 999–1028 (1997)

    MathSciNet  MATH  Google Scholar 

  45. Novikov, IYa., Stechkin, S.B.: Basic wavelet theory. Russian Math. Surveys 53(6), 1159–1231 (1998)

    MathSciNet  MATH  Google Scholar 

  46. Ropela, S.: Spline bases in Besov spaces. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 24, 319–325 (1976)

    MathSciNet  MATH  Google Scholar 

  47. Roudenko, S.: Matrix-weighted Besov spaces. Trans. Amer. Math. Soc. 355(1), 273–314 (2002)

    MathSciNet  MATH  Google Scholar 

  48. Rychkov, V.S.: Littlewood-Paley theory and function spaces with \(A_p^{{\rm loc}}\) weights. Math. Nachr. 224, 145–180 (2001)

    MathSciNet  MATH  Google Scholar 

  49. Schmeisser, H.-J., Triebel, H.: Topics in Fourier Analysis and Function Spaces. Willey, New York (1987)

    MATH  Google Scholar 

  50. Schott, T.: Function spaces with exponential weights. I. Math. Nahr. 189, 221–242 (1998)

    MathSciNet  MATH  Google Scholar 

  51. Schott, T.: Function spaces with exponential weights. II. Math. Nahr. 196, 231–250 (1998)

    MathSciNet  MATH  Google Scholar 

  52. Selected Works of S.L. Sobolev. Volume I: Equations of mathematical physics, computational mathematics, and cubature formulas, Springer, New York (2006)

  53. Sickel, W., Triebel, H.: Hölder inequalities and sharp embeddings in function spaces of \(B_{pq}^s\) and \(F_{pq}^s\) type. Z. Anal. Anwendungen 14, 105–140 (1995)

    MathSciNet  MATH  Google Scholar 

  54. Stein, E.M.: Harmonic Analysis. Princeton University Press, Princeton (1993)

    Google Scholar 

  55. Sjölin, P., Strómberg, J.-O.: Spline systems as bases in Hardy spaces. Israel J. Math. 45(2–3), 147–156 (1983)

    MathSciNet  MATH  Google Scholar 

  56. Strómberg, J.-O., Torchinsky, A.: Weighted Hardy Spaces, Lecture Notes in Math., Vol. 1381, Springer, Berlin (1989)

    MATH  Google Scholar 

  57. Torchinsky, A.: Real-Variable Methods in Harmonic Analysis, Pure and Applied Mathematics, vol. 123. Academic Press Inc., Orlando (1986)

    MATH  Google Scholar 

  58. Triebel, H.: Spline bases and spline representations in function spaces. Arch. Math. 36, 348–359 (1981)

    MathSciNet  MATH  Google Scholar 

  59. Triebel, H.: Theory of Function Spaces. Birkhäuser Verlag, Basel (1983)

    MATH  Google Scholar 

  60. Triebel, H.: Theory of Function Spaces II. Birkhäuser Verlag, Basel (1992)

    MATH  Google Scholar 

  61. Triebel, H.: Fractals and Spectra: Related to Fourier Analysis and Function Spaces. Birkhäuser, Basel (1997)

    MATH  Google Scholar 

  62. Triebel, H.: Theory of Function Spaces III. Birkhäuser Verlag, Basel (2006)

    MATH  Google Scholar 

  63. Triebel, H.: Local means and wavelets in function spaces. In: Function spaces VIII. Vol. 7. Banach Center Publications, pp. 215–234 (2008)

  64. Triebel, H.: Bases in function spaces, sampling, discrepancy, numerical integration, European Math. Soc. Publishing House, Zurich (2010)

  65. Ushakova, E.P., Ushakova, K.E.: Localisation property of Battle–Lemarié wavelets’ sums. J. Math. Anal. Appl. 461(1), 176–197 (2018)

    MathSciNet  MATH  Google Scholar 

  66. Wojciechowska, A.: Local means and wavelets in function spaces with local Muckenhoupt weights. In: Function spaces IX. Vol. 92. Banach Center Publications, pp. 399–412 (2011)

  67. Wojciechowska, A.: Multidimensional wavelet bases in Besov and Triebel–Lizorkin spaces, PhD. thesis, Adam Mickiewicz University Poznań, Poznań (2012)

  68. Wojtaszczyk, P.: A Mathematical Introduction to Wavelets. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

Download references

Acknowledgements

Author gratefully acknowledges the reviewer for careful reading of the manuscript, the new information provided, valuable suggestions and constructive remarks which helped to correct some inaccuracies and significantly improve presentation of the material.

Author information

Authors and Affiliations

  1. Computing Center of the Far-Eastern Branch of the Russian Academy of Sciences, Kim Yu Chena 65, Khabarovsk, Russia, 680000

    Elena P. Ushakova

Authors
  1. Elena P. Ushakova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elena P. Ushakova.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The research was partially supported by the Russian Foundation for Basic Research (Project 19-01-00223).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ushakova, E.P. Spline wavelet bases in function spaces with Muckenhoupt weights. Rev Mat Complut 33, 125–160 (2020). https://doi.org/10.1007/s13163-019-00306-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13163-019-00306-1

Keywords

Mathematics Subject Classification

Search

Navigation