Geodesic
On the recovery of the coefficients of a rearranged Haar series
Sbornik. Mathematics, Tome 58 (1987) no. 1, pp. 31-57 
G. M. Mushegyan. On the recovery of the coefficients of a rearranged Haar series. Sbornik. Mathematics, Tome 58 (1987) no. 1, pp. 31-57. http://geodesic.mathdoc.fr/item/SM_1987_58_1_a2/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The author has previously established that, for any rearrangement of a Haar series, if the rearranged series converges everywhere on $[0,1]$ to a bounded function $f(x)$ then the coefficients of the series can be recovered from the sum $f(x)$ using the formulas of Fourier; however, no analogous assertion holds, in general, for an arbitrary summable function $f$. Generalizing his previous results, the author shows, in particular, that the theorem on the recovery of the coefficients of a rearranged Haar series goes over to functions of the class $L_2$, but not to the class $L_p$ for any $p<2$. Bibliography: 9 titles.

[1] Arutyunyan F. G., “O ryadakh po sisteme Khaara”, DAN ArmSSR, 42 (1966), 134–140 | Zbl

[2] Arutyunyan F. G., Talalyan A. A., “O edinstvennosti ryadov po sistemam Khaara i Uolsha”, Izv. AN SSSR. Ser. matem., 28 (1964), 1391–1408 | Zbl

[3] Mushegyan G. M., “O edinstvennosti ryadov po perestavlennym sistemam Khaara”, Izv. AN ArmSSR. Ser. matem., VI (1971), 21–34

[4] Mushegyan G. M., “O koeffitsientakh vsyudu skhodyaschikhsya ryadov po nekotorym predstavlennym ortonormirovannym sistemam”, Izv. AN SSSR. Ser. matem., 42 (1978), 807–832 | MR | Zbl

[5] Mushegyan G. M., “O koeffitsientakh vsyudu skhodyaschikhsya ryadov po sisteme Khaara s perestavlennymi chlenami”, Izv. AN ArmSSR. Ser. matem., XIII (1978), 275–300 | MR

[6] Petrovskaya M. B., “Nekotorye teoremy edinstvennosti dlya ryadov po sisteme Khaara”, Vestn. MGU, ser. 1, 1964, 15–28

[7] Skvortsov V. A., “Vychislenie koeffitsientov vsyudu skhodyaschegosya ryada Khaara”, Matem. sb., 75(117) (1968), 349–360 | Zbl

[8] Faber G., “Uber die Orthogonal funktionen des Herrn Haar”, Jahrebericht Deutschen Math. Verlinigung, 19 (1910), 104–112 | Zbl

[9] Haar Alfred, Gesamelte Arbeiten, Budapest, 1959