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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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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/

[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