Approximation of functions by the Fourier-Walsh sums (Q580616)

The author proves the following theorem: Let X be the space \(L^ 1[0,1]\) or \(L^{\infty}[0,1]\). Then for every natural number \[ \sup_{f\in X_{\epsilon}}\| f-S_ N(f)\|_ x\epsilon_ N+\sum^{N}_{n=1}\epsilon_{N+n}\int^{1/n}_{1/(n+1)}| D_ N(x)| dx, \] where \(D_ N(x)\) is the Dirichlet-Walsh kernel. In the trigonometric case equivalent results were obtained by the author and K. I. Oskolkov.