mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, moduli of smoothness are used to quantitatively measure smoothness of functions. Moduli of smoothness generalise

modulus of continuity In mathematical analysis, a modulus of continuity is a function ω : , ∞→ , ∞used to measure quantitatively the uniform continuity of functions. So, a function ''f'' : ''I'' → R admits ω as a modulus of continuity if and only if :, f(x)-f ...

and are used in

approximation theory In mathematics, approximation theory is concerned with how function (mathematics), functions can best be approximation, approximated with simpler functions, and with quantitative property, quantitatively characterization (mathematics), characteri ...

and

numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...

to estimate errors of approximation by

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

s and splines.

Moduli of smoothness

The modulus of smoothness of order

n

DeVore, Ronald A., Lorentz, George G., Constructive approximation, Springer-Verlag, 1993. of a function

f\in C,b /math> is the function \omega_n:,\infty)\to\R defined by

: \omega_n(t,f, \Delta_h^n(f,x) \right "> \qquad \text \quad 0\le t\le \frac n, and

: \omega_n(t,f,,b =\omega_n\left(\frac,f,[a,bright) \qquad \text \quad t>\frac, where the finite difference (''n''-th order forward difference) is defined as

: \Delta_h^n(f,x_0)=\sum_^n(-1)^\binom f(x_0+ih).

Properties

\omega_n(0)=0, \omega_n(0+)=0.

\omega_n

is non-decreasing on

[0,\infty).

\omega_n

is continuous on

[0,\infty).

4. For

m\in\N, t\geq 0

we have: ::

\omega_n(mt)\leq m^n\omega_n(t).

\omega_n(f,\lambda t)\leq (\lambda +1)^n\omega_n(f,t),

for

\lambda>0.

6. For

r\in \N

let

W^r

denote the space of continuous function on

[-1,1]

that have

(r-1)

-st absolutely continuous derivative on

[-1,1]

and ::

\left \, f^ \right \, _<+\infty.

:If

f\in W^r,

then ::

\leq t^r \left \, f^ \right \, _, t\geq 0,

:where

\, g(x)\, _=_, 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 In mathematics, the Whitney inequality gives an upper bound for the error of best approximation of a function by polynomials in terms of the moduli of smoothness. It was first proved by Hassler Whitney in 1957, and is an important tool in the fiel ...

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

2\pi

-periodic continuous function, there exists a

trigonometric polynomial In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more natural numbers. The c ...

T_n

of degree

\le n

such that :

\left , f(x)-T_n(x \right ), \leq c(k)\omega_k\left(\frac,f\right),\quad x\in,2\pi

where the constant

c(k)

depends on

k\in\N.

References

{{reflist Approximation theory Numerical analysis