Abstract
In this paper we propose the generalized pseudo-Butterworth refinable functions which involve pseudo-splines of type I and II, Butterworth refinable functions, pseudo-Butterworth refinable functions, and almost all symmetric and causal fractional B-splines. Furthermore, the convergence of cascade algorithms associated with the new masks is proved, and Riesz wavelet bases in L 2(ℝ) corresponding to the parameters are constructed. The regularity of the generalized pseudo-Butterworth refinable functions is also analyzed by Fourier analysis.
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References
Blu, T., Unser, M.: The fractional spline wavelet transform: definition and implementation. In: IEEE Int. Conf. Acoust., Speech, and Sig. Processing, vol. 1, pp. 512–515 (2000)
Daubechies, I.: Ten Lectures on Wavelets, 1 edn. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1992)
Daubechies, I., Han, B., Ron, A., Shen, Z.W.: Framelets: MRA-based constructions of wavelet frames. Appl. Comput. Harmon. Anal. 14, 1–46 (2003)
Deslauriers, G., Dubuc, S.: Symmetric iterative interpolation processes. Constr. Approx. 5, 49–68 (1989)
Dong, B., Shen, Z.W.: Construction of biorthogonal wavelets from pseudo-splines. J. Approx. Theory 138, 211–231 (2006)
Dong, B., Shen, Z.W.: Linear independence of pseudo-splines. Proc. Am. Math. Soc. 134, 2685–2694 (2006)
Dong, B., Shen, Z.W.: Pseudo-splines, wavelets and framelets. Appl. Comput. Harmon. Anal. 22, 78–104 (2007)
Dong, B., Dyn, N., Hormann, K.: Properties of dual pseudo-splines. Appl. Comput. Harmon. Anal. 29, 104–110 (2010)
Dyn, N., Hormann, K., Sabin, M.A., Shen, Z.W.: Polynomial reproduction by symmetric subdivision schemes. J. Approx. Theory 155, 28–42 (2008)
Han, B.: Vector cascade algorithms and refinable function vectors in Sobolev spaces. J. Approx. Theory 124, 44–88 (2003)
Han, B.: Refinable functions and cascade algorithms in weighted spaces with Hölder continuous masks. SIAM J. Math. Anal. 40, 70–102 (2008)
Hormann, K., Sabin, M.A.: A family of subdivision schemes with cubic precision. Comput. Aided Geom. Des. 25, 41–52 (2008)
Kim, H.O., Kim, R.Y.: Sobolev exponents of Butterworth refinable functions. Appl. Math. Lett. 21, 510–515 (2008)
Kim, H.O., Kim, R.Y., Kim, S.S.: Pseudo-Butterworth refinable functions. Curr. Dev. Theory Appl. Wavelets 4, 1–38 (2010)
Kittipoom, P., Kutyniok, G., Lim, W.Q.: Construction of compactly supported shearlet frames. Constr. Approx. 35, 21–72 (2012)
Li, S., Shen, Y.: Shearlet frames with short support. arXiv:1101.4725 (2011)
Shen, Y., Li, S.: Cascade algorithm associated with Hölder continuous masks. Appl. Math. Lett. 22, 1213–1216 (2009)
Shen, Y., Li, S.: Wavelets and framelets from dual pseudo splines. Sci. China Ser. A 54, 1–10 (2011)
Shen, Y., Li, S., Mo, Q.: Complex wavelets and framelets from pseudo splines. J. Fourier Anal. Appl. 16, 885–900 (2010)
Unser, M., Blu, T.: Fractional splines and wavelets. SIAM Rev. 42, 43–67 (2000)
Zhuang, X.: Construction of symmetric complex tight wavelet frames from pseudo splines via matrix extension with symmetry. arXiv:1003.3500 (2010)
Acknowledgements
The authors would like to thank the referees for their valuable comments and suggestions that have improved the paper immeasurably. The authors are grateful to the support of the National Natural Science Foundation of China (Nos. 61033012, 10801023, 10911140268 and 11171052) and Specialized Research Fund of the Doctoral Program of Higher Education of China (No. 20100041110036).
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Communicated by Yang Wang.
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Luo, Z., Qi, W., Zhou, M. et al. Generalized Pseudo-Butterworth Refinable Functions. J Fourier Anal Appl 19, 296–311 (2013). https://doi.org/10.1007/s00041-012-9250-5
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DOI: https://doi.org/10.1007/s00041-012-9250-5