Modulus of smoothness: Difference between revisions

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===Moduli of smoothness===

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Modulus of smoothness of order n of a function f∈C[a,b] is the function <math>\omega_n</math>:[0,∞)→ℝ 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(\frac{b-a}{n},f,[a,b]),</math> for <math>t>\frac{b-a}{n}.</math>

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.

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

===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>

Here we used the definition of the [[https://en.wikipedia.org/wiki/Finite_difference|\fined difference]] (n-th order forward difference)

Modulus of smoothness of order n of a function f∈C<math>f\in C[a,b]</math> is the function <math>\omega_n</math>:[0,∞\infty)→ℝ\to\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 | \qquad \text{for} \quad 0\le t\le \frac{b-a} n,</math>

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

and

===Properties:===

1. :<math>\omega_n(0t,f,[a,b])=0,</math> <math>\omega_n\left(0+\frac{b-a}{n},f,[a,b]\right)=0. \qquad \text{for} \quad t>\frac{b-a}{n},</math>

where the [[finite difference]] (''n''-th order forward difference) is 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, \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>

Here:where <math>\|g(x)\|_{L_{\infty}[-1,1]}={\sup_mathrm{ess} \sup}_{x\in [-1,1]}|g(x)|.</math>

Here <math>\|g(x)\|_{L_{\infty}[-1,1]}=\sup_{x\in [-1,1]}|g(x)|.</math>

==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\N.</math>

==References==

{{reflist}}

[[Category:Approximation theory]]

[[Category:Numerical analysis]]