Geodesic
On the coefficients of Fourier series with respect to the Haar system
Sbornik. Mathematics, Tome 9 (1969) no. 1, pp. 93-109 
S. V. Bochkarev. On the coefficients of Fourier series with respect to the Haar system. Sbornik. Mathematics, Tome 9 (1969) no. 1, pp. 93-109. http://geodesic.mathdoc.fr/item/SM_1969_9_1_a3/
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Voir la notice de l'article provenant de la source Math-Net.Ru

If a continuous function is expanded according to a complete orthonormal system of discontinuous functions (such as the Walsh and Haar systems), the expansion coefficients are subject not only to upper bounds but to lower bounds as well. In this paper lower bounds are derived for the Haar–Fourier coefficients, valid for different classes of continuous functions, and a connection is established between this kind of inequalities and absolute convergence everywhere of the Haar–Fourier series. Bibliography: 7 titles.

[1] N. I. Fine, “On the Walsh functions”, Trans. Amer. Math. Soc., 65 (1949), 372–414 | DOI | MR | Zbl

[2] B. I. Golubov, “O ryadakh Fure nepreryvnykh funktsii po sisteme Khaara”, Izv. AN SSSR, seriya matem., 28 (1964), 1271–1296 | MR | Zbl

[3] P. L. Ulyanov, “O ryadakh po sisteme Khaara”, Matem. sb., 63(105) (1964), 356–391

[4] L. Ciesielski, J. Musielak, “On absolute convergence of Haar series”, Colloq. Math., 7:1 (1959), 61–65 | MR | Zbl

[5] J. R. McLaughlin, “Absolute convergence of Haar series”, Notices Amer. Math. Soc., 14:1 (1967), 102 | MR

[6] H. K. Bari, Trigonometricheskie ryady, Fizmatgiz, M., 1961 | MR

[7] S. Kachmazh, G. Shteingauz, Teoriya ortogonalnykh ryadov, Fizmatgiz, M., 1958 | MR