Abstract:
In this paper, we compute upper bounds for the best approximation by trigonometric polynomials in the metrics $C$ and $L$ on the classes $W^rH_\omega$ of $2\pi$-periodic functions such that $|f^{(r)}(x')-f^{(r)}(x'')|\leqslant\omega(|x'-x''|)$, where $\omega(t)$ is a given convex modulus of continuity. In doing this, we obtain a series of results which explain certain new properties of differentiable functions expressed in terms of rearrangements. Also, we obtain precise estimates for functionals of the form $\int_0^{2\pi}fg\,dx$, where
$f\in H_\omega$, and $g$ belongs to a certain class of differentiable functions defined by bounds on the norm of $g$ and its derivatives in $C$ or $L$.
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