Geodesic
Fourier sums for the Banach indicatrix
Matematičeskie zametki, Tome 15 (1974) no. 4, pp. 527-532

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We prove the existence of a function $f(t)$, which is continuous on the interval $[0,1]$, is of bounded variation, $\min f(t)=0$, $\max f(t)=1$, for which the integral $$ I(x)=\frac1\pi\int_0^\infty\biggl[\int_0^1\cos y(f(t)-x)|df(t)|\biggr]\,dy $$ diverges for almost all $x\in[0,1]$. This result gives a negative answer to a question posed by Z. Ciesielski. 
@article{MZM_1974_15_4_a2,
     author = {K. I. Oskolkov},
     title = {Fourier sums for the {Banach} indicatrix},
     journal = {Matemati\v{c}eskie zametki},
     pages = {527--532},
     publisher = {mathdoc},
     volume = {15},
     number = {4},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_15_4_a2/}
}
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K. I. Oskolkov. Fourier sums for the Banach indicatrix. Matematičeskie zametki, Tome 15 (1974) no. 4, pp. 527-532. http://geodesic.mathdoc.fr/item/MZM_1974_15_4_a2/