Geodesic
On Changes of Variable Preserving the Convergence and Absolute Convergence of Fourier Series in the Haar Wavelet System
Matematičeskie zametki, Tome 108 (2020) no. 2, pp. 179-189.

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It is established that, among all continuously differentiable homeomorphic changes of variable, the absolute convergence of Fourier series in the Haar wavelet system is preserved by only those for which $\varphi^{-1}(0)$ is binary-rational and $\varphi'(x)=\pm 2^m$, where $m$ is an integer and $x\in\mathbb R$. It is also established that this condition is necessary for a continuously differentiable homeomorphic change of variable to preserve the convergence of Fourier series in the Haar wavelet system.
Keywords: Haar wavelets, Fourier–Haar series, continuously differentiable homeomorphism, changes of variable. 
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K. Bitsadze. On Changes of Variable Preserving the Convergence and Absolute Convergence of Fourier Series in the Haar Wavelet System. Matematičeskie zametki, Tome 108 (2020) no. 2, pp. 179-189. http://geodesic.mathdoc.fr/item/MZM_2020_108_2_a2/

[1] G. Alexits, Konvergenzprobleme der Orthogonalreihen, Verlag der Ungarischen Akademie der Wissenschaften, Budapest, 1960 | MR | Zbl

[2] B. S. Kashin, A. A. Saakyan, Ortogonalnye ryady, Nauka, M., 1984 | MR | Zbl

[3] P. L. Ulyanov, “Ob absolyutnoi i ravnomernoi skhodimosti ryadov Fure”, Matem. sb., 72 (114):2 (1967), 193–225 | MR | Zbl

[4] P. L. Ulyanov, “Absolyutnaya skhodimost ryada Fure–Khaara ot superpozitsii funktsii”, Anal. Math., 4:3 (1978), 225–236 | DOI | MR | Zbl

[5] A. M. Olevskii, “Modifikatsiya funktsii i ryady Fure”, UMN, 40:3 (243) (1985), 157–193 | MR | Zbl

[6] V. M. Bugadze, “O ryadakh Fure–Khaara superpozitsii funktsii”, Matem. sb., 182:2 (1991), 175–202 | MR | Zbl

[7] K. R. Bitsadze, “O zamenakh peremennoi, sokhranyayuschikh skhodimost i absolyutnuyu skhodimost ryadov Fure–Khaara”, Matem. sb., 210:6 (2019), 30–55 | DOI | Zbl

[8] K. Bitsadze, “On changes of variable that preserve convergence and absolute convergence of Fourier series with respect to Haar wavelets system”, Bull. Georgian Acad. Sci., 164:3 (2001), 431–432 | MR | Zbl

[9] E. Hernandes, G. Weiss, A First Course on Wavelets, CRC Press, Boca Raton, FL, 1996 | MR

[10] I. P. Natanson, Teoriya funktsii veschestvennoi peremennoi, Nauka, M., 1974 | MR

[11] G. Aleksich, Problemy skhodimosti ortogonalnykh ryadov, IL, M., 1963 | MR | Zbl