Divergent Fourier series with respect to orthonormalized systems of smooth functions

In this paper it is proved that for any functionf L ² –; ], f>0, there exists a complete orthonormalized system of uniformly bounded trigonometric polynomials with respect to which the Fourier series of this function is divergent almost everywhere in the interval –; ].Translated from Matematicheskie Zametki, Vol. 20, No. 1, pp. 69–78, July, 1976.The authoress expresses her gratitude to A. M.