Modulus of smoothness: Difference between revisions

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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 [[polynomial]]s and [[spline (mathematics)|spline]]s.

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===Moduli of smoothness===

The modulus of smoothness of order <math>n</math> <ref>{{cite book |last=DeVore |first=Ronald A. |last2=Lorentz |first2=George G. |title=Constructive approximation |publisher=Springer |series=Grundlehren der mathematischen Wissenschaften |volume=303 |date=1993 |isbn=978-3-642-08075-3 |url=https://link.springer.com/book/9783540506270 |oclc=231539342}}</ref>

The generalization of conception of [[modulus of continuity]] is the modulus of smoothness. If for a contionuous function <math>f</math> the modulus of contuniuty is enough small, then the function is constant. Therefore for higher smoothness we can't use such a criterior as a modulus of continuity. Some properties of a function can be described in terms of a modulus of smoothness.

of a function <math>f\in C[a,b]</math> is the function <math>\omega_n:[0,\infty)\to\mathbb{R}</math> defined by

:<math>\omega_n(t,f,[a,b])=\sup_{h\in[0,t]}\sup_{x\in[a,b-nh]} \left |\Delta_h^n(f,x) \right |</math> \qquad \text{for} <math>t\in[quad 0,\le t\le \frac{b-a}{ n}],</math>

In some problems in the approximation theory the moduli of smoothness are widely used.

Modulus of smoothness of order n

<ref>DeVore, Ronald A., Lorentz, George G., Constructive approximation, Springer-Verlag, 1993.</ref>

of a function <math>f\in C[a,b]</math> is the function <math>\omega_n:[0,\infty)\to\mathbb{R}</math> defined by

:<math>\omega_n(t,f,[a,b])=\sup_{h\in[0,t]}\sup_{x\in[a,b-nh]}|\Delta_h^n(f,x)|</math> for <math>t\in[0,\frac{b-a}{n}],</math>

and

:<math>\omega_n(t,f,[a,b])=\omega_n\left(\frac{b-a}{n},f,[a,b]\right),</math> \qquad \text{for} \quad <math>t>\frac{b-a}{n},</math>

where we the [[finite difference]] (''n''-th order forward difference) areis defined as

:<math>\Delta_h^n(f,x_0)=\sum_{i=10}^n(-1)^{n-i}\binom{n}{i} f(x_0+ih).</math>

===Properties===

1. <math>\omega_n(0)=0,</math> <math>\omega_n(0+)=0.</math>

2. <math>\omega_n</math> is non-decreasing on <math>[0,\infty).</math>

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3. <math>\omega_n</math> is continuous on <math>[0,\infty).</math>

4. <math>\omega_n(mt)\leq m^n\omega_n(t)</math>,For <math>m\in\mathbb{N}</math>, <math>t\geq0.geq 0</math> we have:

::<math>\omega_n(mt)\leq m^n\omega_n(t).</math>

5. <math>\omega_n(f,\lambda t)\leq (\lambda +1)^n\omega_n(f,\lambda t),</math>, for <math>\lambda>0.</math>

6. For <math>r\in \N</math>, denote bylet <math>W^r</math> denote the space of continuous function on <math>[-1,1]</math> that have <math>(r-1)</math>-st absolutely continuous derivative on <math>[-1,1]</math> and
::<math>\left \|f^{(r)} \right \|_{L_{\infty}[-1,1]}<+\infty.</math>
:If <math>f\in W^r,</math>, then
::<math>\omega_r(t,f,[-1,1])\leq t^r \left \|f^{(r)} \right \|_{L_{\infty}[-1,1]}, t\geq 0,</math>
:where <math>\|g(x)\|_{L_{\infty}[-1,1]}={\mathrm{ess} \sup}_{x\in [-1,1]}|g(x)|.</math>

===Application=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 <math>n</math>, if <math>f</math> is <math>2\pi</math>-periodic continuous function, there exists a [[trigonometric polynomial]] <math>T_n</math> of degree <math>\le n</math> such that

:<math>\left |f(x)-T_n(x \right )|\leq c(k)\omega_k\left(\frac{1}{n},f\right),\quad x\in[0,2\pi],</math>

where the constant <math>c(k)</math> depends on <math>k\in\mathbb{N}.</math>

==References==

where the constant <math>c(k)</math> depends on <math>k\in\mathbb{N}.</math>

{{reflist}}

[[Category:Approximation theory]]

== Mathematical analysis / Moduli of smoothness ==

[[Category:Numerical analysis]]