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\addcontentsline{toc}{section}{Ivanov~V.F., Petrov~I.I. The Best Approximation of Periodic Functions by Splines}
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{\Large\bf The Best Approximation of Periodic Functions by Splines
}
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%*********************** AUTHORS ****************************
\centerline{\large V.F.~Ivanov, I.I.~Petrov}

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We establish new theorem on correlation between the sharp constants in the Kolmogorov type inequalities
and the sharp constants in the Kolmogorov-Remez type inequalities
for differentiable periodic functions.
As a special case, we prove the new sharp Kolmogorov-Remez type inequalities for such functions.
We prove also new sharp Bernstein-Remez type inequalities for trigonometric polynomials and polynomial splines.

Let $I_{2 \pi}$ be the unite circle which is realized as interval with coincident endpoint.
Denote by $\varphi_{r}$ the $2\pi$-periodic spline of Euler of order $r$ and let
$$
L(x)_p:=\sup \left\{\left \|x \right\|_{L_p[a, b]}: \; [a, b]\subset I_{2\pi},\; |x(t)|>0,\;t\in (a, b) \right\}.
$$

Our main result are the following theorems.

\textbf{Theorem 1.} 
\textit{Let $k,r \in {\rm \bf N},\;k<r$; $q, p \ge 1$,  $q \ge
rp/(r-k)$; $\alpha = (r-k+1/q)/(r+1/p)$; $ \beta \in [0, 2\pi)$.
If the inequality
$$
\|x^{(k)}\|_{L_q(I_{2\pi})} \leq \frac{ \|\varphi _{r-k}\|_{L_q(I_{2\pi})} } {\|\varphi
_r\|_{L_p(I_{2\pi})}^{\alpha}}\left\|x\right\| _{L_p(I_{2\pi})}^{\alpha} \left\| x^{\left(
r\right) }\right\| _{\infty }^{1-\alpha} \eqno(1)
$$
holds true on the set $L_{\infty}^{r}(I_{2 \pi})$
with $q=\overline{q}$, $p=\overline{p}$ and $\alpha = \overline{\alpha}=(r-k+1/\overline{q})/(r+1/\overline{p})$,
then the following sharp inequality holds
$$
\|x^{(k)}\|_{L_q(I_{2\pi})} \le \|\varphi_r\|_{L_q(I_{2\pi})}
\left\{\frac{\|x\|_{L_p(I_{2\pi}\setminus
B)}}{\|\varphi_r\|_{L_p(I_{2\pi}\setminus
B_1)}}\right\}^{\alpha}\|x^{(r)}\|^{1-\alpha}_\infty  \eqno(2)
$$
for any $ q \ge \overline{q}$, $p\in (0, \overline{p}]$, for arbitrary function
$x \in L_{\infty}^{r}(I_{2 \pi})$ satisfying
$
L(x)_p\le 2^{-\frac1p}\; \|x\|_{L_p(I_{2\pi})}\,,
$
}
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\small{ Dniprovski National\\ University \\ e-mail: \
Ivanov.VF@\@gmail.com }
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\small{ Dniprovski National\\ University \\  e-mail: \
Petrov@\@i.ua 
}
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