For generalized Poisson integrals with kernel parameter r in (0,1), the uniform deviation of Fourier sums is bounded above by a sharp constant times the L_p best approximation error of the generalized derivative, for all 1≤p<∞.
Nikol’skii, Approximation of functions in the mean by trigonometrical polynomials, (in Russian) Izv
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Asymptotically best possible Lebesque-type inequalities for the Fourier sums on sets of generalized Poisson integrals
For generalized Poisson integrals with kernel parameter r in (0,1), the uniform deviation of Fourier sums is bounded above by a sharp constant times the L_p best approximation error of the generalized derivative, for all 1≤p<∞.