Modulus of smoothness

Moduli of smoothness

The modulus of smoothness of order n of a function f ∈ C [ a , b ] is the function ω n : [ 0 , ∞ ) → R defined by

and

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

Properties

1. ω n ( 0 ) = 0 , ω n ( 0 + ) = 0.

2. ω n is non-decreasing on [ 0 , ∞ ) .

3. ω n is continuous on [ 0 , ∞ ) .

4. ω n ( m t ) ≤ m n ω n ( t ) , m ∈ N , t ≥ 0.

5. ω n ( f , λ t ) ≤ ( λ + 1 ) n ω n ( f , t ) , λ > 0.

6. For r ∈ N , denote by W r the space of continuous function on [ − 1 , 1 ] that have ( r − 1 ) -st absolutely continuous derivative on [ − 1 , 1 ] and ∥ f ( r ) ∥ L ∞ [ − 1 , 1 ] < + ∞ . If f ∈ W r , then ω r ( t , f , [ − 1 , 1 ] ) ≤ t r ∥ f ( r ) ∥ L ∞ [ − 1 , 1 ] , t ≥ 0 , where ∥ g ( x ) ∥ L ∞ [ − 1 , 1 ] = e s s sup x ∈ [ − 1 , 1 ] | g ( x ) | .

Applications

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 n , if f is 2 π -periodic continuous function, there exists a trigonometric polynomial T n of degree ≤ n such that

where the constant c ( k ) depends on k ∈ N .