Inequivalence of Wavelet Systems in L₁ (R^d) and BV(R^d)

• Autorzy: Bechler P.

Identyfikatory

DOI

10.4064/ba53-1-4

• Języki publikacji: EN

Abstrakty

EN

Theorems stating sufficient conditions for the inequivalence of the d-variate Haar wavelet system and another wavelet system in the spaces L₁ (R^d) and BV(R^d) are proved. These results are used to show that the Strömberg wavelet system and the system of continuous Daubechies wavelets with minimal supports are not equivalent to the Haar system in these spaces. A theorem stating that some systems of smooth Daubechies wavelets are not equivalent to the Haar system in L₁ (R^d) is also shown.

Słowa kluczowe

EN

Haar system; wavelets; inequivalence

PL

układ Haara; falki; nierównoważność

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2005
• Tom: Vol. 53, no 1
• Strony: 25--37
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Bechler P.

• Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, P.O. Box 21, 00-956 Warszawa, Poland,

Bibliografia

• [1] P. Bechler, R. DeVore, A. Kamont, G. Petrova, and P. Wojtaszczyk, Greedy wavelet projections are bounded in BV, Trans. Amer. Math. Soc., to appear.
• [2] Z. Ciesielski, Properties of the orthonormal Franklin system, Studia Math. 23 (1963), 141-157.
• [3] A. Cohen, W. Dahmen, I. Daubechies, and R. DeVore, Harmonic analysis of the space BV, Rev. Mat. Iberoamericana 19 (2003), 235-263.
• [4] A. Cohen, R. DeVore, P. Petrushev, and H. Xu, Nonlinear approximation and the space BV(R2), Amer. J. Math. 121 (1999), 587-628.
• [5] I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), 909-996.
• [6] —, Ten Lectures on Wavelets, CBMS-NSF Reg. Conf. Ser. Appl. Math. 61, SIAM, Philadelphia, PA, 1992.
• [7] S. Jaffard, Beyond Besov spaces. II. Oscillation spaces, Constr. Approx. 21 (2005), 29-61.
• [8] Y. Meyer, Wavelets and Operators, Cambridge Stud. Adv. Math. 37, Cambridge Univ. Press, Cambridge, 1992, Translated from the 1990 French original by D. H. Salinger.
• [9] D. Pollen, Daubechies’ scaling function on [0, 3], in: Wavelets, Wavelet Anal. Appl. 2, Academic Press, Boston, MA, 1992, 3-13.
• [10] P. Sjölin, The Haar and Franklin systems are not equivalent bases in L1, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 25 (1977), 1099-1100.
• [11] J.-O. Strömberg, A modified Franklin system and higher-order spline systems on Rn as unconditional bases for Hardy spaces, in: Conference on Harmonic Analysis in Honor of Antoni Zygmund, Vol. I, II (Chicago, IL, 1981), Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, 475-494.
• [12] P. Wojtaszczyk, A Mathematical Introduction to Wavelets, London Math. Soci. Student Texts 37, Cambridge Univ. Press, Cambridge, 1997.
• [13] P. Wojtaszczyk, Projections and non-linear approximation in the space BV(Rd), Proc. London Math. Soc. (3) 87 (2003), 471-497.
• [14] W. P. Ziemer, Weakly Differentiable Functions, Grad. Texts in Math. 120, Springer, New York, 1989.