On a certain problem of Lusin (Q2567421)

In the paper the problem of Luzin is solved on the determination of the maximal in the absolute value Fourier coefficients of a given function \(f\in L_2[0,2\pi].\) The method is based on the use of iterated convolutions \(f^{(n+1)}(x) = (f\ast f^{(n)})(x),\) in which the maximal Fourier coefficients play a decisive role. The author obtains the explicit formulae for the values and numbers of the maximal Fourier coefficients, containing the passing to the limit when the number of convolutions \(n \to \infty.\) The formulae of numbers of the maximal Fourier coefficients turn out to be the integer values a priori, so that the approximated calculations together with an ordinary rounding-off error give the absolutely exact result.