View source for Favard problem

The problem of calculating the least upper bound

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038290/f0382901.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038290/f0382902.png" /> are trigonometric polynomials of order not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038290/f0382903.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038290/f0382904.png" /> is the class of periodic functions whose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038290/f0382905.png" />-th derivative in the sense of Weyl (see [[Fractional integration and differentiation|Fractional integration and differentiation]]) satisfies the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038290/f0382906.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038290/f0382907.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038290/f0382908.png" /> and arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f038/f038290/f0382909.png" /> 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>