Geodesic
Divergent Fourier series of integrable functions with respect to orthonormal systems
Matematičeskie zametki, Tome 63 (1998) no. 5, pp. 717-724

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There exist two orthonormal systems such that the Fourier series of each function $f\in L[0,1]$, $f\ne0$, with respect to at least one of these systems diverges on a set of positive measure. 
@article{MZM_1998_63_5_a9,
     author = {A. S. Krantsberg},
     title = {Divergent {Fourier} series of integrable functions with respect to orthonormal systems},
     journal = {Matemati\v{c}eskie zametki},
     pages = {717--724},
     publisher = {mathdoc},
     volume = {63},
     number = {5},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1998_63_5_a9/}
}
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A. S. Krantsberg. Divergent Fourier series of integrable functions with respect to orthonormal systems. Matematičeskie zametki, Tome 63 (1998) no. 5, pp. 717-724. http://geodesic.mathdoc.fr/item/MZM_1998_63_5_a9/