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The problem of calculating the least upper bound
 
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>
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$$ \tag{* }
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\sup _ {\begin{array}{c}
 +
{} \\
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f \in W  ^ {r} MX
 +
\end{array}
 +
} \
 +
\inf _ {t _ {n} } \
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\| f ( x) - t _ {n} ( x) \| _ {X} ,
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$$
  
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]]]).
+
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====
 
====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>
 
<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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 19:38, 5 June 2020


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) satisfies the inequality $ \| f ^ { ( r) } \| _ {X} \leq M $, and $ X = C [ 0, 2 \pi ] $. The Favard problem was posed by J. Favard [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 [2], [3]).

References

[1] 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
[2] 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)
[3] 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)
[4] N.P. Korneichuk, "Extremal problems in approximation theory" , Moscow (1976) (In Russian)

Comments

References

[a1] R.P. Feinerman, D.J. Newman, "Polynomial approximation" , Williams & Wilkins pp. Chapt. IV.4
How to Cite This Entry:
Favard problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Favard_problem&oldid=46907
This article was adapted from an original article by Yu.N. Subbotin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article