Exact errors of best approximation for complex-valued periodic functions

We extend Nagy's theorem on best approximation by trigonometric polynomials in the $L_1$ metric to certain complex-valued periodic functions. We use this result to find exact constants of best approximation in $L_1$ and $L_\infty$ on some complex convolution classes. For classes of real-valued convolutions these constants were found by Nikol'skii. As an example, we apply these results to the Schwarz kernel and to the corresponding convolution classes.
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