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General Wavelet Frames in \( L^{2}(\mathbb{R}) \)

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An Introduction to Frames and Riesz Bases

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

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Abstract

A fundamental question in wavelet analysis is what conditions we have to impose on a function ψ such that a given signal \( f \in L^{2}(\mathbb{R}) \) can be expanded via translated and scaled versions of ψ, i.e., via functions

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References

  1. Bownik, M., Lemvig, J.: The canonical and alternate duals of a wavelet frame. Appl. Comput. Harmon. Anal. 23, 263–272 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Christensen, O., Goh, S.S.: From dual pairs of Gabor frames to dual pairs of wavelet frames and vice versa. Appl. Comput. Harmon. Anal. 46, 198–214 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chui, C., Shi, X.: Bessel sequences and affine frames. Appl. Comput. Harmon. Anal. 1, 29–49 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chui, C., Shi, X.: Inequalities of Littlewood-Paley type for frames and wavelets. SIAM J. Math. Anal. 24(1), 263–277 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chui, C., Shi, X.: Orthonormal wavelets and tight frames with arbitrary real dilations. Appl. Comput. Harmon. Anal. 9(3), 243–264 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Daubechies, I.: The wavelet transformation, time-frequency localization and signal analysis. IEEE Trans. Inf. Theory 36, 961–1005 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)

    Book  MATH  Google Scholar 

  8. Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27, 1271–1283 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  9. Daubechies, I., Han, B.: The canonical dual of a wavelet frame. Appl. Comput. Harmon. Anal. 12(3), 269–285 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Grossmann, A., Morlet, J.: Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J. Math. Anal. 15, 723–736 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. Heil, C., Walnut, D.: Continuous and discrete wavelet transforms. SIAM Rev. 31, 628–666 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  12. Holighaus, N., Wiesmeyr, C.: Construction of warped time-frequency representations on nonuniform frequency scales, Part I: Frames (2015, preprint)

    Google Scholar 

  13. Holschneider, M., Tchamitchiam, P.: Régularité locale de la fonction “nondifférentiable” de Riemann. In: Lemarié, P.G. (ed.) Les ondelettes en 1989. Lecture Notes in Mathematics, vol. 1438. Springer, New York (1989)

    Google Scholar 

  14. Laugesen, R.S.: On affine frames with transcendental dilations. Proc. Am. Math. Soc. 135, 211–216 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lemvig, J.: Constructing pairs of dual bandlimited framelets with desired time localization. Adv. Comput. Math. 30, 231–247 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Christensen, O. (2016). General Wavelet Frames in \( L^{2}(\mathbb{R}) \) . In: An Introduction to Frames and Riesz Bases. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-25613-9_15

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