Abstract
In this paper, we present a construction of frame without using the Fourier transform. Our methods are based on the Calderón–Zygmund operator theory and Coifman’s decomposition of the identity operator, which also work on homogeneous spaces in the sense of Coifman and Weiss.
Dedicated to Professor John Benedetto on his eightieth birthday.
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References
Bremer, J., Coifman, R., Maggioni, M., and Szlam, A.: Diffusion wavelet packets. Appl. Comput. Harmon. Anal. 21 95–112 (2006)
Bui, H. and Laugesen R.S.: Wavelet frame bijectivity on Lebesgue and Hardy spaces. J. Fourier Anal. Appl. 19 376–409 (2013)
Calderón, A.: Intermediate spaces and interpolation, the complex method. Studia Math. 24 113–190 (1964)
Coifman, R.: A real variable characterization of Hp. Studia Math. 51 269–274 (1974)
Coifman, R.: Wavelet analysis and signal processing. In: Signal Processing, Part I, in: IMA Vol. Math. Appl., vol. 22, pp. 59–68. Springer, New York (1990)
Coifman, R. and Maggioni, M.: Diffusion wavelets. Appl. Comput. Harmon. Anal. 21 (1) 53–94 (2006)
Coifman, R. and Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Amer. Math. Soc. 83 569–645 (1977)
David, G., Journé, J.-L., and Semmes, S.: Calderón-Zygmund operators, para-accretive functions and interpolation. Rev. Mat. Iberoamericana 1 1–56 (1985)
Deng, D. and Han, Y.: Harmonic analysis on spaces of homogeneous type. Lecture Notes in Math., vol. 1966. Springer-Verlag, Berlin (2009)
Ding, Y., Han, Y., Lu, G., and Wu, X.: Boundedness of singular integrals on multiparameter weighted Hardy spaces \(H^p_w(\mathbb {R}^n\times \mathbb {R}^m)\). Potential Anal. 37 31–56 (2012)
Frazier, M. and Jawerth, B.: A discrete transform and decomposition of distribution spaces. J. Funct. Anal. 93 34–170 (1990)
Han, Y.: Calderón-type reproducing formula and the Tb theorem. Rev. Mat. Iberoamericana 10 51–91 (1994)
Meyer, Y.: Wavelets and operators. Cambridge studies in advanced mathematics 37. Cambridge University Press (1992)
Taibleson, M. and Weiss, G.: The molecular characterization of certain Hardy spaces. Astérisque 77 67–149 (1980)
Acknowledgements
The first author is partially supported by an NSF grant DMS-1408839 and a McDevitt Endowment Fund at Georgetown University, and the third author partially supported by NNSF of China (Grant No. 11671397) and Special Funds for Yueqi Young Scholars of CUMTB (2017QN29).
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Chang, DC., Han, Y., Wu, X. (2021). Construction of Frames Using Calderón–Zygmund Operator Theory. In: Hirn, M., Li, S., Okoudjou, K.A., Saliani, S., Yilmaz, Ö. (eds) Excursions in Harmonic Analysis, Volume 6. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-69637-5_8
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DOI: https://doi.org/10.1007/978-3-030-69637-5_8
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