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The problem of calculating the least upper bound

$$ \tag{* }
\sup _ {\begin{array}{c}
{} \\
 f \in W  ^ {r} MX 
\end{array}
 } \ 
\inf _ {t _ {n} } \ 
\| f ( x) - t _ {n} ( x) \| _ {X} ,
$$

where the  $  t _ {n} ( x) $
are trigonometric polynomials of order not exceeding  $  n $, 
$  W  ^ {r} MX $
is the class of periodic functions whose  $  r $-
th derivative in the sense of Weyl (see [[Fractional integration and differentiation|Fractional integration and differentiation]]) satisfies the inequality  $  \| f ^ { ( r) } \| _ {X} \leq  M $, 
and  $  X = C [ 0, 2 \pi ] $. 
The Favard problem was posed by J. Favard [[#References|[1]]]. Subsequently, broader classes of functions have been considered and a complete solution of the Favard problem for  $  X = C, L $
and arbitrary  $  r > 0 $
has been obtained as a corollary of more general results (see [[#References|[2]]], [[#References|[3]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Favard,   "Sur les meilleurs procédés d'approximation de certaines classes de fonctions par des polynômes trigonométriques"  ''Bull. Sci. Math.'' , '''61'''  (1937)  pp. 209–224</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S.B. Stechkin,   "On best approximation of certain classes of periodic functions by trigonometric functions"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''20''' :  5  (1956)  pp. 643–648  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.K. Dzyadyk,   "Best approximation on classes of periodic functions defined by kernels which are integrals of absolutely monotone functions"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''23''' :  6  (1959)  pp. 933–950  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.P. Korneichuk,   "Extremal problems in approximation theory" , Moscow  (1976)  (In Russian)</TD></TR></table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.P. Feinerman,   D.J. Newman,   "Polynomial approximation" , Williams &amp; Wilkins  pp. Chapt. IV.4</TD></TR></table>