We developed a continuous wavelet theory on nonlinear spectral groups induced by frequency warpings. Let $ \varphi $: $ I \to \mathbb{R} $ be a real $ \mathcal{C}^1 $-diffeomorphism and define
$ (\mathcal{F}_\varphi f)(t) = \widehat{f}(\varphi(t)), \quad d\mu_\varphi(t) = \frac{|\varphi'(t)|}{2\pi}\, dt. $
The transform $ \mathcal{F}_\varphi $ is unitary from $ L^2(\mathbb{R}) $ onto $ L^2(I, d\mu_\varphi) $, and it diagonalizes the warped spectral operator
$ T_\varphi = i\varphi^{-1}(D), \quad D = -i\partial_x. $
The change of variables $ \xi = \varphi(t) $ transports the additive Fourier group to the nonlinear spectral group
$ t\oplus_\varphi s = \varphi^{-1}(\varphi(t)+\varphi(s)). $
Using the corresponding warped shifts and automorphic dilations, we constructed continuous warped-shift wavelets and proved their Calderón admissibility, Plancherel, and reconstruction formulas. The theory becomes the classical continuous wavelet transform after linearization by $ \varphi $, while in the variable $ t $, it yields a nonlinear spectral wavelet analysis. We also separated this construction from Fourier-adapted and operator-adapted warped wavelets, and explained its connection with finite operator calculus, umbral generating functions, and the Meixner–Pollaczek central-difference operator. Examples include logarithmic, exponential, sine, tangent, hyperbolic tangent, and hyperbolic-sine warpings.
Citation: Fethi Bouzeffour, Mhamed Eddahbi. Continuous warped-shift wavelets on nonlinear spectral groups[J]. AIMS Mathematics, 2026, 11(8): 23792-23816. doi: 10.3934/math.2026958
We developed a continuous wavelet theory on nonlinear spectral groups induced by frequency warpings. Let $ \varphi $: $ I \to \mathbb{R} $ be a real $ \mathcal{C}^1 $-diffeomorphism and define
$ (\mathcal{F}_\varphi f)(t) = \widehat{f}(\varphi(t)), \quad d\mu_\varphi(t) = \frac{|\varphi'(t)|}{2\pi}\, dt. $
The transform $ \mathcal{F}_\varphi $ is unitary from $ L^2(\mathbb{R}) $ onto $ L^2(I, d\mu_\varphi) $, and it diagonalizes the warped spectral operator
$ T_\varphi = i\varphi^{-1}(D), \quad D = -i\partial_x. $
The change of variables $ \xi = \varphi(t) $ transports the additive Fourier group to the nonlinear spectral group
$ t\oplus_\varphi s = \varphi^{-1}(\varphi(t)+\varphi(s)). $
Using the corresponding warped shifts and automorphic dilations, we constructed continuous warped-shift wavelets and proved their Calderón admissibility, Plancherel, and reconstruction formulas. The theory becomes the classical continuous wavelet transform after linearization by $ \varphi $, while in the variable $ t $, it yields a nonlinear spectral wavelet analysis. We also separated this construction from Fourier-adapted and operator-adapted warped wavelets, and explained its connection with finite operator calculus, umbral generating functions, and the Meixner–Pollaczek central-difference operator. Examples include logarithmic, exponential, sine, tangent, hyperbolic tangent, and hyperbolic-sine warpings.
| [1] |
A. P. Calderón, Intermediate spaces and interpolation, the complex method, Stud. Math., 24 (1964), 113–190. https://doi.org/10.4064/SM-24-2-113-190 doi: 10.4064/SM-24-2-113-190
|
| [2] |
J. Morlet, G. Arens, E. Fourgeau, D. Giard, Wave propagation and sampling theory; Part Ⅰ, complex signal and scattering in multilayered media, Geophysics, 47 (1982), 203–221. https://doi.org/10.1190/1.1441328 doi: 10.1190/1.1441328
|
| [3] |
A. Grossmann, J. Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. Math. Anal., 15 (1984), 723–736. https://doi.org/10.1137/0515056 doi: 10.1137/0515056
|
| [4] | A. Grossmann, J. Morlet, T. Paul, Transforms associated to square integrable group representations Ⅱ: examples, Ann. Inst. Henri Poincaré, 45 (1986), 293–309. |
| [5] |
I. Daubechies, Orthonormal bases of compactly supported wavelets, Commun. Pure Appl. Math., 41 (1988), 909–996. https://doi.org/10.1002/cpa.3160410705 doi: 10.1002/cpa.3160410705
|
| [6] | I. Daubechies, Ten lectures on wavelets, Philadelphia, 1992. https://doi.org/10.1137/1.9781611970104 |
| [7] |
S. Mallat, A theory for multiresolution signal decomposition: the wavelet representation, IEEE Trans. Pattern Anal. Mach. Intell., 11 (1989), 674–693. https://doi.org/10.1109/34.192463 doi: 10.1109/34.192463
|
| [8] | S. Mallat, A wavelet tour of signal processing, 2 Eds., Academic Press, 1999. |
| [9] | P. P. Vaidyanathan, Multirate systems and Filter banks, Prentice Hall, 1993. |
| [10] | M. Vetterli, J. Kovačević, Wavelets and subband coding, Prentice Hall, 1995. |
| [11] |
A. V. Oppenheim, D. H. Johnson, K. Steiglitz, Computation of spectra with unequal resolution using the fast Fourier transform, Proc. IEEE, 59 (1971), 299–301. https://doi.org/10.1109/PROC.1971.8146 doi: 10.1109/PROC.1971.8146
|
| [12] |
C. Braccini, A. V. Oppenheim, Unequal bandwidth spectral analysis using digital frequency warping, IEEE Trans. Acoust. Speech Signal Process., 22 (1974), 236–244. https://doi.org/10.1109/TASSP.1974.1162582 doi: 10.1109/TASSP.1974.1162582
|
| [13] |
R. G. Baraniuk, D. L. Jones, Unitary equivalence: a new twist on signal processing, IEEE Trans. Signal Process., 43 (1995), 2269–2282. https://doi.org/10.1109/78.469861 doi: 10.1109/78.469861
|
| [14] |
G. Evangelista, S. Cavaliere, Frequency-warped filter banks and wavelet transforms: a discrete-time approach via Laguerre expansion, IEEE Trans. Signal Process., 46 (1998), 2638–2650. https://doi.org/10.1109/78.720367 doi: 10.1109/78.720367
|
| [15] |
G. Evangelista, Dyadic warped wavelets, Adv. Imaging Electron. Phys., 117 (2001), 73–170. https://doi.org/10.1016/S1076-5670(01)80112-9 doi: 10.1016/S1076-5670(01)80112-9
|
| [16] | G. C. Rota, D. Kahaner, A. Odlyzko, Finite operator calculus, J. Math. Anal. Appl., 42 (1973), 684–760. |
| [17] |
D. Zeilberger, Some comments on Rota's umbral calculus, J. Math. Anal. Appl., 74 (1980), 456–463. https://doi.org/10.1016/0022-247X(80)90141-9 doi: 10.1016/0022-247X(80)90141-9
|
| [18] |
J. Shen, Combinatorics for wavelets: the umbral refinement equation, Stud. Appl. Math., 103 (1999), 121–147. https://doi.org/10.1111/1467-9590.00123 doi: 10.1111/1467-9590.00123
|
| [19] |
M. E. H. Ismail, N. Saad, On the Meixner–Pollaczek polynomials and the Sturm–Liouville problems, J. Math. Anal. Appl., 552 (2025), 129794. https://doi.org/10.1016/j.jmaa.2025.129794 doi: 10.1016/j.jmaa.2025.129794
|
Fethi Bouzeffour, Mhamed Eddahbi. Continuous warped-shift wavelets on nonlinear spectral groups[J]. AIMS Mathematics, 2026, 11(8): 23792-23816. doi: 10.3934/math.2026958
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