Geodesic
Logarithmic growth of the $L^1$-norm of the majorant of partial sums of an orthogonal series
Matematičeskie zametki, Tome 58 (1995) no. 2, pp. 218-230.

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It is proved that for any $N\times N$ orthogonal matrix $A=\{a_{ij}\}$ we have $$ \sum_{i=1}^N\max_{1\le n\le N}\biggl|\sum_{j=1}^na_{ij}\biggr| \ge\frac 1{30}N^{1/2}\log N. $$ A multidimensional analog of this result is also established. 
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B. S. Kashin; S. I. Sharek. Logarithmic growth of the $L^1$-norm of the majorant of partial sums of an orthogonal series. Matematičeskie zametki, Tome 58 (1995) no. 2, pp. 218-230. http://geodesic.mathdoc.fr/item/MZM_1995_58_2_a4/

[1] Olevskii A. M., Fourier series with respect to General Orthogonal Systems, Springer, Berlin, 1975

[2] Kashin B. S., Saakyan A. A., Ortogonalnye ryady, Nauka, M., 1984 | Zbl

[3] Stechkin S. B., “Ob absolyutnoi skhodimosti ortogonalnykh ryadov”, DAN SSSR, 102:1 (1955), 37–40 | MR | Zbl

[4] Khardi G., Littlvud D., Polia G., Neravenstva, IL, M., 1948

[5] Stechkin S. B., “O nailuchshikh priblizheniyakh zadannykh klassov funktsii lyubymi polinomami”, Uspekhi matem. nauk, 9:1 (1954), 133–134 | Zbl

[6] Ismagilov R. S., “Ob $n$-mernykh poperechnikakh kompaktov v gilbertovom prostranstve”, Funkts. anal. i ego prilozh., 2:2 (1968), 32–39 | MR | Zbl

[7] Temlyakov V. N., Priblizhenie funktsii s ogranichennoi smeshannoi proizvodnoi, Tr. MIAN, 178, Nauka, M., 1986 | MR | Zbl