Geodesic
An example of an orthonormal system of convergence in $C$ but not in $L^2$
Sbornik. Mathematics, Tome 20 (1973) no. 1, pp. 145-153 
A. M. Olevskii. An example of an orthonormal system of convergence in $C$ but not in $L^2$. Sbornik. Mathematics, Tome 20 (1973) no. 1, pp. 145-153. http://geodesic.mathdoc.fr/item/SM_1973_20_1_a7/
@article{SM_1973_20_1_a7,
     author = {A. M. Olevskii},
     title = {An~example of an orthonormal system of convergence in~$C$ but not in~$L^2$},
     journal = {Sbornik. Mathematics},
     pages = {145--153},
     year = {1973},
     volume = {20},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1973_20_1_a7/}
}
TY  - JOUR
AU  - A. M. Olevskii
TI  - An example of an orthonormal system of convergence in $C$ but not in $L^2$
JO  - Sbornik. Mathematics
PY  - 1973
SP  - 145
EP  - 153
VL  - 20
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1973_20_1_a7/
LA  - en
ID  - SM_1973_20_1_a7
ER  - 
%0 Journal Article
%A A. M. Olevskii
%T An example of an orthonormal system of convergence in $C$ but not in $L^2$
%J Sbornik. Mathematics
%D 1973
%P 145-153
%V 20
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1973_20_1_a7/
%G en
%F SM_1973_20_1_a7

Voir la notice de l'article provenant de la source Math-Net.Ru

We prove the following theorem. Theorem. {\it For any $p_0\in[1,\infty)$ there exists a complete uniformly bounded orthonormal system $\{\varphi_n\}$ having the following properties}: 1) For all $f\in L^p, p>p_0,$ the Fouries series $\sum c_n\varphi_n$ converges to $f$ almost everywhere. 2) {\it There exists an $F\in L^{p_0}$ whose Fourier series diverges almost everywhere.} Bibliography: 8 titles.

[1] A. M. Olevskii, “Raskhodyaschiesya ryady Fure ot nepreryvnykh funktsii”, DAN SSSR, 141:1 (1961), 28–31 | MR | Zbl

[2] S. Kachmazh, G. Shteingauz, Teoriya ortogonalnykh ryadov, Fizmatgiz, Moskva, 1958

[3] P. L. Ulyanov, “Reshennye i nereshennye problemy teorii trigonometricheskikh i ortogonalnykh ryadov”, Uspekhi matem. nauk, XIX:1 (127) (1964), 3–69 | MR

[4] A. Zigmund, Trigonometricheskie ryady, t. 2, izd-vo «Mir», Moskva, 1965 | MR

[5] A. M. Olevskii, “Raskhodyaschiesya ryady iz $L^2$ po polnym sistemam”, DAN SSSR, 138:3 (1961), 545–548 | MR | Zbl

[6] P. L. Ulyanov, “Raskhodyaschiesya ryady po sisteme Khaara i po bazisam”, DAN SSSR, 138:3 (1961), 556–559

[7] E. M. Nikishin, P. L. Ulyanov, “Ob absolyutnoi i bezuslovnoi skhodimosti”, Uspekhi matem. nauk, XXII:3 (135) (1965), 240–242

[8] A. M. Olevskii, “Ob odnoi ortonormalnoi sisteme i ee primeneniyakh”, Matem. sb., 71 (113) (1966), 297–336 | MR | Zbl