Modulus of smoothness


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In mathematics, moduli of smoothness are used to quantitatively measure smoothness of functions. Moduli of smoothness generalise modulus of continuity and are used in approximation theory and numerical analysis to estimate errors of approximation by polynomials and splines.

Moduli of smoothnessEdit

The modulus of smoothness of order   [1] of a function   is the function   defined by




where the finite difference (n-th order forward difference) is defined as




2.   is non-decreasing on  

3.   is continuous on  

4. For   we have:


5.   for  

6. For   let   denote the space of continuous function on   that have  -st absolutely continuous derivative on   and

If   then


Moduli of smoothness can be used to prove estimates on the error of approximation. Due to property (6), moduli of smoothness provide more general estimates than the estimates in terms of derivatives.

For example, moduli of smoothness are used in Whitney inequality to estimate the error of local polynomial approximation. Another application is given by the following more general version of Jackson inequality:

For every natural number  , if   is  -periodic continuous function, there exists a trigonometric polynomial   of degree   such that


where the constant   depends on  


  1. ^ DeVore, Ronald A., Lorentz, George G., Constructive approximation, Springer-Verlag, 1993.