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Band-limited wavelets beyond Gevrey regularity

Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Trg Dositeja Obradovića 4, Novi Sad 21000, Serbia

E-mail Address: nenad.teofanov@dmi.uns.ac.rs

Corresponding author.

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Filip Tomić

https://orcid.org/0000-0003-2453-6769

Faculty of Technical Sciences, University of Novi Sad, Trg Dositeja Obradovića 6, Novi Sad 21000, Serbia

E-mail Address: filip.tomic@uns.ac.rs

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, and

Stefan Tutić

https://orcid.org/0000-0002-4229-3644

Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Trg Dositeja Obradovića 4, Novi Sad 21000, Serbia

E-mail Address: stefan.tutic@dmi.uns.ac.rs

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https://doi.org/10.1142/S0219691324500243Cited by:0 (Source: Crossref)

Abstract

It is known that a smooth function of exponential decay at infinity cannot be an orthonormal wavelet. Dziubański and Hernández constructed smooth orthonormal wavelets of Gevrey-type subexponential decay. We weaken the Gevrey-type decay and construct orthonormal wavelets of subexponential decay related to the so-called extended Gevrey classes. The virtue of our construction is that precise asymptotics of functions from such classes can be given in terms of the Lambert $W$ $W$ -function.

Keywords:

AMSC: 42C40, 46E10, 46F05

References

1. P. Auscher, G. Weiss and M. V. Wickerhauser , Local sine and cosine basis of Coifman and Meyer and the construction of smooth wavelets, in Wavelets: A Tutorial in Theory and Applications, ed. C. K. Chui (Academic Press, 1992), pp. 237–256. Crossref, Google Scholar
2. J.-B. Burie, A. Ducrot and Q. Griette , Asymptotic behavior of an epidemic model with infinitely many variants, J. Math. Biol. 87 (2023) 40. Crossref, Web of Science, Google Scholar
3. M. Cicognani and D. Lorentz , Strictly hyperbolic equations with coefficients low-regular in time and smooth in space, J. Pseudo-Differ. Oper. Appl. 9(3) (2018) 643–675. Crossref, Web of Science, Google Scholar
4. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth , On the Lambert $W$ $W$ function, Adv. Comput. Math. 5 (1996) 329–359. Crossref, Web of Science, Google Scholar
5. I. Daubechies , Ten Lectures on Wavelets (SIAM, Philadelphia, 1992). Crossref, Google Scholar
6. J. Dziubański and E. Hernández , Band-limited wavelets with subexponential decay, Can. Math. Bull. 41 (1998) 398–403. Crossref, Web of Science, Google Scholar
7. N. Fukuda, T. Kinoshita and I. Uehara , On the wavelets having Gevrey regularities and subexponential decays, Math. Nachr. 287 (2014) 546–560. Crossref, Web of Science, Google Scholar
8. S. Fürdös, D. N. Nenning, A. Rainer and G. Schindl , Almost analytic extensions of ultradifferentiable functions with applications to microlocal analysis, J. Math. Anal. Appl. 481(1) (2020) 123451. Crossref, Web of Science, Google Scholar
9. I. M. Gelfand and G. E. Shilov , Generalized Functions II (Academic Press, New York, 1968). Google Scholar
10. E. Hernández and G. Weiss , A First Course of Wavelets (CRC Press, 1996). Crossref, Google Scholar
11. A. Hoorfar and M. Hassani , Inequalities on the Lambert $W$ $W$ function and hyperpower function, J. Inequal. Pure Appl. Math. 9 (2008) 51. Google Scholar
12. L. Hörmander , The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis (Springer, 1990). Crossref, Google Scholar
13. A. Israel and A. Mayeli , On the eigenvalue distribution of spatio-spectral limiting operators in higher dimensions, Appl. Comput. Harmon. Anal. 70 (2024) 101620. Crossref, Web of Science, Google Scholar
14. J. Jiménez-Garrido, A. Lastra and J. Sanz , Extension operators for some ultraholomorphic classes defined by sequences of rapid growth, Constr. Approx. (2023), https://doi.org/10.1007/s00365-023-09663-z. Crossref, Web of Science, Google Scholar
15. J. Jiménez-Garrido, D. N. Nenning and G. Schindl , On generalized definitions of ultradifferentiable classes, J. Math. Anal. Appl. 526 (2023) 127260. Crossref, Web of Science, Google Scholar
16. H. Komatsu , Ultradistributions, I: Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo, Sect. I A, Math. 20(1) (1973) 25–105. Google Scholar
17. E. A. Lebedeva , Minimization of uncertainty constant for a family of Meyer wavelets, Math. Notes 81 (2007) 489–495. Crossref, Google Scholar
18. E. A. Lebedeva , On the uncertainty principle for Meyer wavelet functions, J. Math. Sci. 182 (2012) 656–662. Crossref, Google Scholar
19. P. G. Lemarié and Y. Meyer , Ondelettes et bases Hilbertiennes, Rev. Mat. Iberoam. 2 (1986) 1–18. Crossref, Google Scholar
20. S. Mandelbrojt , Séries adherents: Régularisation des suites: Applications (Gauthier-Villars, Paris, 1952). Google Scholar
21. Y. Meyer , Principe d’incertitude, bases Hilbertiennes et algébres d’opérateurs, Sémin. Bourbaki 28 (1985–1986) 209–223. Google Scholar
22. I. Mező , The Lambert W Function: Its Generalizations and Applications (CRC Press, Boca Raton, 2022). Crossref, Google Scholar
23. S. Moritoh and K. Tomoeda , A further decay estimate for the Dziubański-Hernández wavelets, Can. Math. Bull. 53 (2010) 133–139. Crossref, Web of Science, Google Scholar
24. E. Pigani, D. Sgarbossa, S. Suweis, A. Maritan and S. Azaele , Delay effects on the stability of large ecosystems, Proc. Natl. Acad. Sci. USA 119 (2022) e2211449119. Crossref, Web of Science, Google Scholar
25. S. Pilipović, D. Rakić, N. Teofanov and J. Vindas , Multiresolution expansions and wavelets in Gelfand–Shilov spaces, RACSAM 114 (2020) 66. https://doi.org/10.1007/s13398-020-00789-4. Crossref, Google Scholar
26. S. Pilipović, N. Teofanov and F. Tomić , On a class of ultradifferentiable functions, Novi Sad J. Math. 45(1) (2015) 125–142. Crossref, Google Scholar
27. S. Pilipović, N. Teofanov and F. Tomić , Superposition and propagation of singularities for extended Gevrey regularity, FILOMAT 32 (2018) 2763–2782. Crossref, Google Scholar
28. S. Pilipović, N. Teofanov and F. Tomić , A Paley–Wiener theorem in extended Gevrey regularity, J. Pseudo-Differ. Oper. Appl. 11 (2020) 593–612. Crossref, Web of Science, Google Scholar
29. A. Rainer and G. Schindl , Composition in ultradifferentiable classes, Stud. Math. 224 (2014) 97–131. Crossref, Web of Science, Google Scholar
30. L. Rodino , Linear Partial Differential Operators in Gevrey Spaces (World Scientific, 1993). Link, Google Scholar
31. H.-G. Roos, L. Teofanov and Z. Uzelac , A modified Bakhvalov mesh, Appl. Math. Lett. 31 (2014) 7–11. Crossref, Web of Science, Google Scholar
32. N. Teofanov and F. Tomić , Inverse closedness and singular support in extended Gevrey regularity, J. Pseudo-Differ. Oper. Appl. 8 (2017) 411–421. Crossref, Google Scholar
33. N. Teofanov and F. Tomić , Extended Gevrey regularity via weight matrices, Axioms 11 (2022) 576. Crossref, Web of Science, Google Scholar
34. Y. Yang , Electromagnetic asymmetry, relegation of curvature singularities of charged black holes, and cosmological equations of state in view of the Born-Infeld theory, Class. Quantum Grav. 39 (2022) 195007. Crossref, Web of Science, Google Scholar