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The modulus of continuity of the derivative of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s0859201.png" /> of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s0859202.png" /> defined on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s0859203.png" />, i.e. the expression
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<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/s/s085/s085920/s0859204.png" /></td> </tr></table>
+
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 +
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<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/s/s085/s085920/s0859205.png" /></td> </tr></table>
+
The modulus of continuity of the derivative of order  $  m \geq  1 $
 +
of a function  $  f $
 +
defined on a Banach space  $  X $,
 +
i.e. the expression
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s0859206.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s0859207.png" />, the modulus of smoothness is the ordinary modulus of continuity (cf. [[Continuity, modulus of|Continuity, modulus of]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s0859208.png" />. Basic properties of the modulus of smoothness (in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s0859209.png" /> and the space of continuous functions) are:
+
$$
 +
\omega _ {m} ( f, \delta , X) =
 +
$$
  
<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/s/s085/s085920/s08592010.png" /></td> </tr></table>
+
$$
 +
= \
 +
\sup _ {\begin{array}{c}
 +
h,x \in X \\
 +
\| h \| _ {X} \leq  \delta
 +
\end{array}
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592011.png" /> does not decrease together with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592012.png" />;
+
}  \left \| \sum _ {i = 0 } ^ { m }  (- 1) ^ {m - i } \left ( \begin{array}{c}
 +
m \\
 +
i
 +
\end{array}
 +
\right ) f \left ( x + ( m - 2i) {
 +
\frac{h}{2}
 +
} \right ) \right \| _ {X} ,
 +
$$
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592013.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592014.png" /> is an integer, then
+
where  $  ( x \pm  mh/2) \in X $.  
 +
If  $  m = 1 $,
 +
the modulus of smoothness is the ordinary modulus of continuity (cf. [[Continuity, modulus of|Continuity, modulus of]]) of  $  f $.  
 +
Basic properties of the modulus of smoothness (in the case  $  X = \mathbf C $
 +
and the space of continuous functions) are:
  
<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/s/s085/s085920/s08592015.png" /></td> </tr></table>
+
$$
 +
\omega _ {m} ( f, 0, \mathbf C )  = 0;
 +
$$
  
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592016.png" />,
+
$  \omega _ {m} ( f, \delta , \mathbf C ) $
 +
does not decrease together with  $  \delta $;
  
<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/s/s085/s085920/s08592017.png" /></td> </tr></table>
+
if  $  k $
 +
$  \geq  1 $
 +
is an integer, then
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592018.png" />, then
+
$$
 +
\omega _ {m} ( f, k \delta , \mathbf C )  \leq  \
 +
k  ^ {m} \omega _ {m} ( f, \delta , \mathbf C );
 +
$$
  
<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/s/s085/s085920/s08592019.png" /></td> </tr></table>
+
for any  $  \lambda > 0 $,
  
if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592020.png" />, then
+
$$
 +
\omega _ {m} ( f, \lambda \delta , \mathbf C )  \leq  \
 +
( \lambda + 1)  ^ {m} \omega _ {m} ( f, \delta , \mathbf C );
 +
$$
  
<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/s/s085/s085920/s08592021.png" /></td> </tr></table>
+
if  $  \nu > m $,
 +
then
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592023.png" /> are constants independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592024.png" />.
+
$$
 +
\omega _  \nu  ( f, \delta , \mathbf C )  \leq  \
 +
2 ^ {\nu - m } \omega _ {m} ( f, \delta , \mathbf C );
 +
$$
  
Certain problems in the theory of approximation of functions can ultimately be solved only in terms of a modulus of smoothness of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592025.png" />. In the theory of approximations of functions an important class is the class of periodic continuous functions with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592026.png" /> and with second-order modulus of smoothness satisfying the condition
+
if  $  \nu > m $,
 +
then
  
<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/s/s085/s085920/s08592027.png" /></td> </tr></table>
+
$$
 +
\omega _ {m} ( f, \delta , \mathbf C )
 +
\leq  A _ {\nu , m }  \delta  ^  \nu
 +
\int\limits _  \delta  ^ { a }
 +
 
 +
\frac{\omega _ {m} ( f, u , \mathbf C ) }{u ^ {\nu + 1 } }
 +
\
 +
du + O ( \delta  ^  \nu  ),
 +
$$
 +
 
 +
where  $  A _ {\nu , m }  $
 +
and  $  a $
 +
are constants independent of  $  f $.
 +
 
 +
Certain problems in the theory of approximation of functions can ultimately be solved only in terms of a modulus of smoothness of order  $  m \geq  2 $.
 +
In the theory of approximations of functions an important class is the class of periodic continuous functions with period  $  2 \pi $
 +
and with second-order modulus of smoothness satisfying the condition
 +
 
 +
$$
 +
\omega _ {2} ( f, \delta , C _ {2 \pi }  )  \leq  \delta .
 +
$$
  
 
The modulus of continuity of such functions satisfies the condition
 
The modulus of continuity of such functions satisfies the condition
  
<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/s/s085/s085920/s08592028.png" /></td> </tr></table>
+
$$
 +
\omega _ {1} ( f, \delta , C _ {2 \pi }  )  \leq  \
 +
\left [
 +
\frac{1}{ \mathop{\rm ln}  ( \sqrt {2 } + 1) }
 +
\right ]
 +
\delta  \mathop{\rm ln}  {
 +
\frac \pi  \delta
 +
} + O ( \delta ),
 +
$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592029.png" />, and the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592030.png" /> cannot be improved [[#References|[4]]].
+
$  0 < \delta \leq  \pi $,  
 +
and the constant $  1/ \mathop{\rm ln}  ( \sqrt {2 } + 1) $
 +
cannot be improved [[#References|[4]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.N. Bernshtein,  "Sur l'ordre de la meilleure approximation des fonctions continués par de polynomes de degré donné"  ''Mem. Publ. Classe Sci. Acad. Belgique (2)'' , '''4'''  (1912)  pp. 1–103</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Marchaud,  "Sur les dérivées et sur les différences des fonctions de variables réelles"  ''J. Math. Pures Appl.'' , '''6'''  (1927)  pp. 337–425</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Zygmund,  "Smooth functions"  ''Duke Math. J.'' , '''12'''  (1945)  pp. 47–76</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.V. Efimov,  "Estimate of the modules of continuity of a function in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592031.png" />"  ''Izv. Akad. Nauk SSSR Ser Mat.'' , '''21'''  (1957)  pp. 283–288  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.N. Bernshtein,  "Sur l'ordre de la meilleure approximation des fonctions continués par de polynomes de degré donné"  ''Mem. Publ. Classe Sci. Acad. Belgique (2)'' , '''4'''  (1912)  pp. 1–103</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Marchaud,  "Sur les dérivées et sur les différences des fonctions de variables réelles"  ''J. Math. Pures Appl.'' , '''6'''  (1927)  pp. 337–425</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Zygmund,  "Smooth functions"  ''Duke Math. J.'' , '''12'''  (1945)  pp. 47–76</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.V. Efimov,  "Estimate of the modules of continuity of a function in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592031.png" />"  ''Izv. Akad. Nauk SSSR Ser Mat.'' , '''21'''  (1957)  pp. 283–288  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The modulus of smoothness <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592032.png" /> is also written in terms of symmetric differences, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592033.png" />, where
+
The modulus of smoothness $  \omega _ {m} ( f, \delta ) $
 +
is also written in terms of symmetric differences, as $  \omega _ {m} ( f , \delta ) = \sup _ {0 < h \leq  \delta }  \| \Delta _ {h}  ^ {m} f \| $,  
 +
where
  
<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/s/s085/s085920/s08592034.png" /></td> </tr></table>
+
$$
 +
\Delta _ {h}  ^ {1} f ( x)  = \
 +
f \left ( x +
 +
\frac{h}{2}
 +
\right ) - f \left ( x -  
 +
\frac{h}{2}
 +
\right )
 +
$$
  
 
and
 
and
  
<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/s/s085/s085920/s08592035.png" /></td> </tr></table>
+
$$
 +
\Delta _ {h}  ^ {m} f ( x)  = \
 +
\Delta _ {n} ( \Delta _ {n}  ^ {m-} 1 f( x))  = \
 +
\sum _ { i= } 0 ^ { m }  (- 1)  ^ {m-} i \left ( \begin{array}{c}
 +
m \\
 +
i
 +
\end{array}
 +
\right )
 +
f \left ( x + ( m- 2i)
 +
\frac{h}{2}
 +
\right ) .
 +
$$
  
 
This gives a recurrent procedure for computing (approximations of) it.
 
This gives a recurrent procedure for computing (approximations of) it.
  
To overcome certain shortcomings of this (classical) modulus of smoothness (especially its ability to characterize the order of the best polynomial approximation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592036.png" /> to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592037.png" />), new moduli of smoothness have been introduced. They use so-called step-weight functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592038.png" />, and are defined by
+
To overcome certain shortcomings of this (classical) modulus of smoothness (especially its ability to characterize the order of the best polynomial approximation $  E _ {n} ( f  ) $
 +
to a function $  f \in L _ {p} [- 1 , 1] $),  
 +
new moduli of smoothness have been introduced. They use so-called step-weight functions $  \varphi ( x) $,  
 +
and are defined by
  
<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/s/s085/s085920/s08592039.png" /></td> </tr></table>
+
$$
 +
\omega _  \varphi  ^ {n} ( f , \delta ) _ {p}  = \
 +
\sup _ {0 < h \leq  \delta } \
 +
\| \Delta _ {h\varphi }  ^ {m} f \| _ {L _ {p}  } .
 +
$$
  
The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592040.png" /> is chosen for the problem at hand. Note that here the increment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592041.png" /> varies with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592042.png" />. A basic result is that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592043.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592044.png" />. (Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592045.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592048.png" />, and approximation is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592049.png" />.) For more on such moduli, their use in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085920/s08592050.png" /> approximation problems and in the interpolation of spaces, see [[#References|[a1]]].
+
The function $  \varphi ( x) $
 +
is chosen for the problem at hand. Note that here the increment $  h \varphi ( x) $
 +
varies with $  x $.  
 +
A basic result is that $  E _ {n} ( f  ) _ {p} = O( n ^ {- \alpha } ) $
 +
if and only if $  \omega _  \varphi  ^ {m} ( f , \delta ) _ {p} = O( t  ^  \alpha  ) $.  
 +
(Here $  0< \alpha < m $,
 +
$  1 \leq  p \leq  \infty $,  
 +
$  \varphi ( x) = ( 1- x  ^ {2} )  ^ {1/2} $,  
 +
$  f \in L _ {p} [- 1, 1] $,  
 +
and approximation is in $  L _ {p} [- 1, 1] $.)  
 +
For more on such moduli, their use in $  L _ {p} $
 +
approximation problems and in the interpolation of spaces, see [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Z. Ditzian,  V. Totik,  "Moduli of smoothness" , Springer  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.G. Lorentz,  "Approximation of functions" , Holt, Rinehart &amp; Winston  (1966)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Z. Ditzian,  V. Totik,  "Moduli of smoothness" , Springer  (1987)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.G. Lorentz,  "Approximation of functions" , Holt, Rinehart &amp; Winston  (1966)</TD></TR></table>

Revision as of 08:14, 6 June 2020


The modulus of continuity of the derivative of order $ m \geq 1 $ of a function $ f $ defined on a Banach space $ X $, i.e. the expression

$$ \omega _ {m} ( f, \delta , X) = $$

$$ = \ \sup _ {\begin{array}{c} h,x \in X \\ \| h \| _ {X} \leq \delta \end{array} } \left \| \sum _ {i = 0 } ^ { m } (- 1) ^ {m - i } \left ( \begin{array}{c} m \\ i \end{array} \right ) f \left ( x + ( m - 2i) { \frac{h}{2} } \right ) \right \| _ {X} , $$

where $ ( x \pm mh/2) \in X $. If $ m = 1 $, the modulus of smoothness is the ordinary modulus of continuity (cf. Continuity, modulus of) of $ f $. Basic properties of the modulus of smoothness (in the case $ X = \mathbf C $ and the space of continuous functions) are:

$$ \omega _ {m} ( f, 0, \mathbf C ) = 0; $$

$ \omega _ {m} ( f, \delta , \mathbf C ) $ does not decrease together with $ \delta $;

if $ k $ $ \geq 1 $ is an integer, then

$$ \omega _ {m} ( f, k \delta , \mathbf C ) \leq \ k ^ {m} \omega _ {m} ( f, \delta , \mathbf C ); $$

for any $ \lambda > 0 $,

$$ \omega _ {m} ( f, \lambda \delta , \mathbf C ) \leq \ ( \lambda + 1) ^ {m} \omega _ {m} ( f, \delta , \mathbf C ); $$

if $ \nu > m $, then

$$ \omega _ \nu ( f, \delta , \mathbf C ) \leq \ 2 ^ {\nu - m } \omega _ {m} ( f, \delta , \mathbf C ); $$

if $ \nu > m $, then

$$ \omega _ {m} ( f, \delta , \mathbf C ) \leq A _ {\nu , m } \delta ^ \nu \int\limits _ \delta ^ { a } \frac{\omega _ {m} ( f, u , \mathbf C ) }{u ^ {\nu + 1 } } \ du + O ( \delta ^ \nu ), $$

where $ A _ {\nu , m } $ and $ a $ are constants independent of $ f $.

Certain problems in the theory of approximation of functions can ultimately be solved only in terms of a modulus of smoothness of order $ m \geq 2 $. In the theory of approximations of functions an important class is the class of periodic continuous functions with period $ 2 \pi $ and with second-order modulus of smoothness satisfying the condition

$$ \omega _ {2} ( f, \delta , C _ {2 \pi } ) \leq \delta . $$

The modulus of continuity of such functions satisfies the condition

$$ \omega _ {1} ( f, \delta , C _ {2 \pi } ) \leq \ \left [ \frac{1}{ \mathop{\rm ln} ( \sqrt {2 } + 1) } \right ] \delta \mathop{\rm ln} { \frac \pi \delta } + O ( \delta ), $$

$ 0 < \delta \leq \pi $, and the constant $ 1/ \mathop{\rm ln} ( \sqrt {2 } + 1) $ cannot be improved [4].

References

[1] S.N. Bernshtein, "Sur l'ordre de la meilleure approximation des fonctions continués par de polynomes de degré donné" Mem. Publ. Classe Sci. Acad. Belgique (2) , 4 (1912) pp. 1–103
[2] A. Marchaud, "Sur les dérivées et sur les différences des fonctions de variables réelles" J. Math. Pures Appl. , 6 (1927) pp. 337–425
[3] A. Zygmund, "Smooth functions" Duke Math. J. , 12 (1945) pp. 47–76
[4] A.V. Efimov, "Estimate of the modules of continuity of a function in the class " Izv. Akad. Nauk SSSR Ser Mat. , 21 (1957) pp. 283–288 (In Russian)

Comments

The modulus of smoothness $ \omega _ {m} ( f, \delta ) $ is also written in terms of symmetric differences, as $ \omega _ {m} ( f , \delta ) = \sup _ {0 < h \leq \delta } \| \Delta _ {h} ^ {m} f \| $, where

$$ \Delta _ {h} ^ {1} f ( x) = \ f \left ( x + \frac{h}{2} \right ) - f \left ( x - \frac{h}{2} \right ) $$

and

$$ \Delta _ {h} ^ {m} f ( x) = \ \Delta _ {n} ( \Delta _ {n} ^ {m-} 1 f( x)) = \ \sum _ { i= } 0 ^ { m } (- 1) ^ {m-} i \left ( \begin{array}{c} m \\ i \end{array} \right ) f \left ( x + ( m- 2i) \frac{h}{2} \right ) . $$

This gives a recurrent procedure for computing (approximations of) it.

To overcome certain shortcomings of this (classical) modulus of smoothness (especially its ability to characterize the order of the best polynomial approximation $ E _ {n} ( f ) $ to a function $ f \in L _ {p} [- 1 , 1] $), new moduli of smoothness have been introduced. They use so-called step-weight functions $ \varphi ( x) $, and are defined by

$$ \omega _ \varphi ^ {n} ( f , \delta ) _ {p} = \ \sup _ {0 < h \leq \delta } \ \| \Delta _ {h\varphi } ^ {m} f \| _ {L _ {p} } . $$

The function $ \varphi ( x) $ is chosen for the problem at hand. Note that here the increment $ h \varphi ( x) $ varies with $ x $. A basic result is that $ E _ {n} ( f ) _ {p} = O( n ^ {- \alpha } ) $ if and only if $ \omega _ \varphi ^ {m} ( f , \delta ) _ {p} = O( t ^ \alpha ) $. (Here $ 0< \alpha < m $, $ 1 \leq p \leq \infty $, $ \varphi ( x) = ( 1- x ^ {2} ) ^ {1/2} $, $ f \in L _ {p} [- 1, 1] $, and approximation is in $ L _ {p} [- 1, 1] $.) For more on such moduli, their use in $ L _ {p} $ approximation problems and in the interpolation of spaces, see [a1].

References

[a1] Z. Ditzian, V. Totik, "Moduli of smoothness" , Springer (1987)
[a2] G.G. Lorentz, "Approximation of functions" , Holt, Rinehart & Winston (1966)
How to Cite This Entry:
Smoothness, modulus of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Smoothness,_modulus_of&oldid=12777
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article