Geodesic
On upper bounds of Fourier--Walsh coefficients
Matematičeskie zametki, Tome 6 (1969) no. 6, pp. 725-736.

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An upper bound is established for the upper bounds of the Fourier–Walsh coefficients $a_n(f)$ whose modulus of continuity $\omega(\delta,f)$ does not exceed a given modulus of continuity $\omega(\delta)$. In the case of convex majorants of $\omega(\delta)$, these bounds are attained for individual ordinal numbers $n$. 
@article{MZM_1969_6_6_a8,
     author = {A. V. Efimov},
     title = {On upper bounds of {Fourier--Walsh} coefficients},
     journal = {Matemati\v{c}eskie zametki},
     pages = {725--736},
     publisher = {mathdoc},
     volume = {6},
     number = {6},
     year = {1969},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1969_6_6_a8/}
}
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A. V. Efimov. On upper bounds of Fourier--Walsh coefficients. Matematičeskie zametki, Tome 6 (1969) no. 6, pp. 725-736. http://geodesic.mathdoc.fr/item/MZM_1969_6_6_a8/