Abstract
Letf be a continuous periodic function with Fourier sums Sn(f), and let En(f)=En be the best approximation tof by trigonometric polynomials of order n. The following estimate is proved: $$||f - S_n (f)|| \leqslant c\sum\nolimits_{v = n}^{2u} {\frac{{E_v }}{{v - n + 1}}} .$$ (Here c is an absolute constant.) This estimate sharpens Lebesgue's classical inequality for “fast” decreasing E v . The sharpness of this estimate is proved for an arbitrary class of functions having a given majorant of best approximations. Also investigated is the sharpness of the corresponding estimate for the rate of convergence of a Fourier series almost everywhere.
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