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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     author = {B. S. Kashin and S. I. Sharek},
     title = {Logarithmic growth of the $L^1$-norm of the majorant of partial sums of an orthogonal series},
     journal = {Matemati\v{c}eskie zametki},
     pages = {218--230},
     publisher = {mathdoc},
     volume = {58},
     number = {2},
     year = {1995},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1995_58_2_a4/}
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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/