# Smoothness, modulus of

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)

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=48741
This article was adapted from an original article by A.V. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article