Uniform-convergence factors for Fourier series of functions with a given modulus of continuity

It is proved that a sequence of factors $\{\lambda_\nu\}$, which are Fourier–Stieitjes coefficients, converts the Fourier series of any function whose modulus of continuity does not exceed a given modulus of continuity $\omega(\delta)$ into a uniformly convergent series, if and only if $\omega(1/n)\int_o^{2\pi}\left|\lambda_0/2+\sum_{\nu=1}^n\lambda_\nu\cos\nu t\right|dt=o(1)$. The sufficiency of this condition is known.