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
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bownik, M., Lemvig, J.: The canonical and alternate duals of a wavelet frame. Appl. Comput. Harmon. Anal. 23, 263–272 (2007)
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)
Chui, C., Shi, X.: Bessel sequences and affine frames. Appl. Comput. Harmon. Anal. 1, 29–49 (1993)
Chui, C., Shi, X.: Inequalities of Littlewood-Paley type for frames and wavelets. SIAM J. Math. Anal. 24(1), 263–277 (1993)
Chui, C., Shi, X.: Orthonormal wavelets and tight frames with arbitrary real dilations. Appl. Comput. Harmon. Anal. 9(3), 243–264 (2000)
Daubechies, I.: The wavelet transformation, time-frequency localization and signal analysis. IEEE Trans. Inf. Theory 36, 961–1005 (1990)
Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)
Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27, 1271–1283 (1986)
Daubechies, I., Han, B.: The canonical dual of a wavelet frame. Appl. Comput. Harmon. Anal. 12(3), 269–285 (2002)
Grossmann, A., Morlet, J.: Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J. Math. Anal. 15, 723–736 (1984)
Heil, C., Walnut, D.: Continuous and discrete wavelet transforms. SIAM Rev. 31, 628–666 (1989)
Holighaus, N., Wiesmeyr, C.: Construction of warped time-frequency representations on nonuniform frequency scales, Part I: Frames (2015, preprint)
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)
Laugesen, R.S.: On affine frames with transcendental dilations. Proc. Am. Math. Soc. 135, 211–216 (2007)
Lemvig, J.: Constructing pairs of dual bandlimited framelets with desired time localization. Adv. Comput. Math. 30, 231–247 (2009)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-319-25613-9_15
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-25611-5
Online ISBN: 978-3-319-25613-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)