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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|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 [[#References|[4]]].

====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>

====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 [[#References|[a1]]].

====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>