Some Smooth Compactly Supported Tight Framelets

A. San Antolín, R. A. Zalik

Abstract


For any dilation matrix with integer entries, we construct a family of smooth compactly supported tight wavelet frames in $L^2(\mathbb{R}^d)$, $d\ge 1$. Estimates for the degrees of smoothness of these framelets are given. Our construction involves some compactly supported refinable functions, the Oblique Extension Principle and a slight generalization of a theorem of Lai and Stöckler.

Keywords


Dilation matrix; Fourier transform; Refinable function; Tight framelet

Full Text:

PDF

References


M. Bownik, Tight frames of multidimensional wavelets, J. Fourier Anal. Appl. 3(5) (1997), 525–542.

C.K. Chui, W. He and J. Stöckler, Compactly supported tight and sibling frames with maximum vanishing moments, Appl. Comput. Harmonic Anal. 13 (2002), 224–262.

I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.

I. Daubechies, B. Han, A. Ron and Z.W. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (2003), 1–46.

K. Gröchening and A. Haas, Self-similar lattice tilings, J. Fourier Analysis and Appl. 1 (1994), 131–170.

K. Gröchening and W.R. Madych, Multiresolution analysis, Haar bases and self-similar tillings of Rn, IEEE Trans. Inform. Theory 38(2) (1992), 556–568.

K. Gröchenig and A. Ron, Tight compactly supported wavelet frames of arbitrarily high smoothness, Proc. Amer. Math. Soc. 126 (1998), 1101–1107.

B. Han, Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix, Approximation Theory, Wavelets and Numerical Analysis, (Chattanooga, TN, 2001), J. Comput. Appl. Math. 155(1) (2003), 43–67.

B. Han, On dual wavelet tight frames, Applied Comput. Harmon. Anal. 4 (1997), 380–413.

E. Hernández and G. Weiss, A First Course on Wavelets, CRC Press Inc., 1996.

M.-J. Lai, Construction of multivariate compactly supported orthonormal wavelets, Adv. Comput. Math. 25 (2006) 41–56.

M.-J. Lai and J. Stöckler, Construction of multivariate compactly supported tight wavelet frames, Appl. Comput. Harmon. Anal. 21 (2006), 324–348.

A. Ron and Z.W. Shen, Affine systems in L2(Rd ): the analysis of the analysis operator, J. Funct. Anal. 148 (1997), 408–447.

A. Ron and Z.W. Shen, Compactly supported tight affine frames in L2(Rd ), Math. Comp. 67 (1998), 191–207.

A. Ron and Z.W. Shen, Construction of compactly supported afine frames in L2(Rd ), in Advances in Wavelets, K.S. Lau (editor), Springer-Verlag, New York, 1998, pp. 27–49.

R.S. Strichartz, Wavelets and self-affine tilings, Constructive Approximation 9(2-3) (1993), 327–346.

P. Wojtaszczyk, A Mathematical Introduction to Wavelets, London Math. Soc., Student Texts 37, 1997.

Y. Shouzhi and X. Yanmei, Construction of compactly supported conjugate symmetric complex tight wavelet frames, Int.J.Wavelets Multiresolut. Inf. Process. 8 (2010), 861–874.




DOI: http://dx.doi.org/10.26713%2Fcma.v3i3.217

Refbacks

  • There are currently no refbacks.


eISSN 0975-8607; pISSN 0976-5905