A decay theorem for refinable functions. (English) Zbl 0851.42027

Summary: We show that a refinable function with absolutely summable mask cannot have exponential decay in both time and frequency.


42C15 General harmonic expansions, frames
Full Text: DOI


[1] Mallat, S., Multiresolution approximation and wavelet orthonormal bases of \(L^2R\), Trans. Am. Math. Soc., 315, 69-87 (1989) · Zbl 0686.42018
[2] Wavelets and Operators (1993), Cambridge University Press: Cambridge University Press London, English transl.
[3] Battle, G., Phase space localization theorem for ondelettes, J. Math. Phys., 30, 2195-2196 (1989) · Zbl 0694.46006
[4] Daubechies, I., Ten Lectures on Wavelets, Number 61 in CBMS-NSF Series in Applied Mathematics (1992), SIAM Publications: SIAM Publications Philadelphia · Zbl 0776.42018
[5] Holschneider, M.; Kronland-Martinet, R.; Morlet, J.; Tchamitchian, Ph., A real-time algorithm for signal analysis with the help of the wavelet transform, (Combes, J. M.; Grossmann, A.; Tchamitchian, Ph., Wavelets: Time-Frequency Methods and Phase Space (1990), Springer-Verlag: Springer-Verlag Berlin), 286-297 · Zbl 0850.94021
[6] Shensa, M., The discrete wavelet transform: Wedding the à trous and Mallat algorithms, IEEE Trans. Signal Process, 40, 2464-2482 (1992) · Zbl 0825.94053
[7] Lemarié, P.-G., Ondelettes à localisation exponentielle, J. Math. Pures et Appl., 67, 227-236 (1988) · Zbl 0758.42020
[8] Daubechies, I., Orthonormal bases of compactly supported wavelets, Comm. Pure and Appl. Math., 41, 909-996 (1988) · Zbl 0644.42026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.