Geodesic
Representation of measurable functions by series in the Faber–Schauder system, and universal series
Izvestiya. Mathematics, Tome 11 (1977) no. 1, pp. 205-218 
V. G. Krotov. Representation of measurable functions by series in the Faber–Schauder system, and universal series. Izvestiya. Mathematics, Tome 11 (1977) no. 1, pp. 205-218. http://geodesic.mathdoc.fr/item/IM2_1977_11_1_a7/
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this work criteria are established for various types of universal series in the Faber–Schauder system of functions. By means of these criteria the maximal speed of decrease is established for coefficients of universal series in this system, and existence is proved for continuous functions with guaranteed (and best possible) smoothness in terms of moduli of continuity whose basis expansions in the Faber–Schauder system are universal in some sense or other. Convergence almost everywhere as well as convergence in integral “metrics” is considered. Bibliography: 11 titles.

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[2] Schauder J., “Zur Theorie stetiger Abbildungen in Funktionenräumen”, Math. Zeit., 26 (1927), 47–65 | DOI | MR | Zbl

[3] Ulyanov P. L., “Predstavlenie izmerimykh funktsii ryadami i klassy $\varphi(L)$”, Uspekhi matem. nauk, 27:2 (1972), 3–52 | MR

[4] Talalyan A. A., “Predstavlenie proizvolnoi izmerimoi funktsii ryadami po funktsiyam sistemy Shaudera”, Izv. AN ArmSSR, ser. fiz.-mat. nauk, 12:3 (1959), 3–12 | MR

[5] Goffman C., “Remark on a problem of Lusin”, Acta Math., 111:1,2 (1964), 63–72 | DOI | MR | Zbl

[6] Zink R., “On a theorem of Goffman concerning Schauder series”, Proc. Amer. Math. Soc., 21:3 (1969), 523–529 | DOI | MR | Zbl

[7] Krotov V. G., “Predstavlenie izmerimykh funktsii ryadami po sisteme Fabera–Shaudera i universalnye ryady”, Dokl. AN SSSR, 214:6 (1974), 1258–1261 | MR | Zbl

[8] Krotov V. G., “Ob universalnykh ryada Fure po sisteme Fabera–Shaudera”, Vestnik MGU, ser. matem. mekh., 1975, no. 4, 53–57 | MR

[9] Krotov V. G., “O ryadakh po sisteme Fabera–Shaudera i po bazisam prostranstva $C[0,1]$”, Matem. zametki, 14:2 (1973), 185–195 | MR | Zbl

[10] Menshov D. E., “O chastnykh summakh trigonometricheskikh ryadov”, Matem. sb., 20:2 (1947), 197–236

[11] Chiesielski Z., “On Haar functions and the Schauder bases of the space $C_{\langle0,1\rangle}$”, Bull. Acad. Polon. Sci., Ser. Math., 7:4 (1959), 227–232 | MR