Geodesic
Approximation of periodic functions by trigonometric polynomials in the mean
Matematičeskie zametki, Tome 16 (1974) no. 1, pp. 15-26.

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For arbitrary summation methods we obtain inequalities between upper bounds of deviations in the $L$ metric and corresponding upper bounds in the $C$ metric with respect to a certain class of functions. These inequalities constitute a generalization of known relationships due to S. M. Nikol'skii. We consider the cases wherein these inequalities become exact or asymptotic equalities. 
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     author = {V. P. Motornyi},
     title = {Approximation of periodic functions by trigonometric polynomials in the mean},
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V. P. Motornyi. Approximation of periodic functions by trigonometric polynomials in the mean. Matematičeskie zametki, Tome 16 (1974) no. 1, pp. 15-26. http://geodesic.mathdoc.fr/item/MZM_1974_16_1_a1/