# 2.2 Sparse approximation and ℓp spaces

 Page 1 / 1

We now look at how well $f\in \mathbf{X}$ can be approximated by $n$ functions in the dictionary $\mathcal{D}$ .

We define the error of $n$ -term approximation of $f$ by the elements of the dictionary $\mathcal{D}$ as $\begin{array}{cc}\text{(1)}& \sigma {}_{n}\left(f\right){}_{\mathbf{X}}:=\sigma {}_{n}\left(f,\mathcal{D}\right){}_{X}:=\text{inf}{}_{s\in {\mathrm{\Sigma }}_{n}}\parallel f-s\parallel {}_{\mathbf{X}}.\end{array}$

We also define the class of $r$ -smooth signals in $\mathcal{D}$ as ,

with the corresponding norm ${\parallel f\parallel }_{{\mathcal{A}}^{r}}={sup}_{n=1,2,\mathrm{\dots }}{\mathcal{n}}^{r}{\sigma }_{n}{\left(f\right)}_{\mathbf{X}}$ .

In general, the larger $r$ is, the 'smoother' the function $s\in {\mathcal{A}}^{r}{\left(\mathcal{D}\right)}_{\mathbf{X}}$ . Note also that ${\mathcal{A}}^{r}\subseteq {\mathcal{A}}^{{r}^{\prime }}$ if $r>{r}^{\prime }$ . Given $f$ , let $r\left(f\right)=\text{sup}\left\{r:f\in {\mathcal{A}}^{r}\right\}$ be a measure of the "smoothness" of $f$ , i.e. a quantification of compressibility.

Let $\mathbf{X}=H$ , a Hilbert space

A Hilbert space is a complete inner product space with the norm induced by the innerproduct
such as $\mathbf{X}={L}_{2}\left(\mathbb{R}\right)$ , and assume $\mathcal{D}=B$ - an orthonormal basis on $\mathbf{X}$ ; i.e. if $B={\left\{{\varphi }_{i}\right\}}_{i}$ , then $〈{\varphi }_{i},{\varphi }_{j}〉={\delta }_{i,j}$ , where ${\delta }_{i,j}$ is the Kronecker delta. This also means that each $f\in \mathbf{X}$ has an expansion $f={\sum }_{j}{c}_{j}\left(f\right){\varphi }_{j}$ , where ${c}_{j}\left(f\right)=〈f,{\varphi }_{j}〉$ . We also have ${\parallel f\parallel }_{\mathbf{X}}^{2}={\sum }_{j=1}^{\mathrm{\infty }}{|{c}_{j}\left(f\right)|}^{2}$ .

Recall the definition of ${\mathrm{\ell }}_{p}$ spaces: let $\left({a}_{j}\right)\in \mathbb{R}$ ; then $\left({a}_{j}\right)\in {\mathrm{\ell }}_{p}$ if ${\parallel \left({a}_{j}\right)\parallel }_{{\mathrm{\ell }}_{p}}<\infty$ with ${\parallel \left({a}_{j}\right)\parallel }_{{\mathrm{\ell }}_{p}}={\left({\sum }_{j}{|{a}_{j}|}^{p}\right)}^{1/p}$ for $p<\mathrm{\infty }$ and ${\parallel \left({a}_{j}\right)\parallel }_{{\mathrm{\ell }}_{p}}={sup}_{j}|{a}_{j}|$ for $p=\mathrm{\infty }$ . We also recall that for ${L}_{p}$ spaces on compact sets, ${L}_{p}\subset {L}_{{p}^{\prime }}$ if $p>{p}^{\prime }$ . The opposite is true for ${\mathrm{\ell }}_{p}$ spaces: ${\mathrm{\ell }}_{p}\subset {\mathrm{\ell }}_{{p}^{\prime }}$ if $p<{p}^{\prime }$ . Hence, the smaller the value of $p$ is, the “smaller” ${\mathrm{\ell }}_{p}$ is.

Does there exist a sequence $\left({a}_{j}\right)$ with ${\parallel \left({a}_{j}\right)\parallel }_{{\mathrm{\ell }}_{1}}={\sum }_{j}|{a}_{j}|<\mathrm{\infty }$ but with ${\parallel \left({a}_{j}\right)\parallel }_{{\mathrm{\ell }}_{p}}={\left({\sum }_{j}{|{a}_{j}|}^{p}\right)}^{\frac{1}{p}}=\mathrm{\infty }$ for all $0 ? Consider the sequence ${a}_{n}=\frac{1}{n{\left(\mathrm{log}n\right)}^{1+\delta }}$ . We see that $\left({a}_{n}\right)\in {\mathrm{\ell }}_{1}$ but ${\parallel \left({a}_{n}\right)\parallel }_{{\mathrm{\ell }}_{p}}=\mathrm{\infty }$ for all $0 .

A sequence $\left({a}_{n}\right)$ is in ${\mathrm{\ell }}_{p}$ if the sorted magnitudes of the ${a}_{n}$ decay faster than ${n}^{\mathrm{-}\frac{1}{p}}$ .

Define ${a}_{n}^{*}$ as the element of the sequence $\left({a}_{n}\right)$ with the ${n}^{\text{th}}$ largest magnitude, and denote $\left({a}_{n}^{*}\right)$ as the decreasing rearrangement of $\left({a}_{n}\right)$ . It is easy to show that $k{\left({a}_{k}^{*}\right)}^{p}\le {\sum }_{n}{\left({a}_{n}\right)}^{p}$ for all $k$ ; also, if $\left({a}_{n}\right)\in {\mathrm{\ell }}_{p}$ , then ${a}_{k}^{*}\le {\parallel \left({a}_{n}\right)\parallel }_{{\mathrm{\ell }}_{p}}{k}^{-\frac{1}{p}}$ .

A sequence $\left({a}_{n}\right)$ is in weak ${\mathrm{\ell }}_{p}$ , denoted $\left({a}_{n}\right)\in w{\mathrm{\ell }}_{p}$ , if ${a}_{k}^{*}\le M{k}^{-\frac{1}{p}}$ . We also define the quasinorm

A quasinorm is has the properties of a norm except thatthe triangle inequality is replaced by the condition $\parallel x+y\parallel \le {C}_{0}\left[\parallel x\parallel +\parallel y\parallel \right]$ for some absolute constant ${C}_{0}$ .
${\parallel \left({a}_{n}\right)\parallel }_{w{\mathrm{\ell }}_{p}}$ as the smallest $M>0$ such that ${a}_{k}^{*}\le M{k}^{-\frac{1}{p}}$ for each $k$ .

The sequence ${a}_{n}=\frac{1}{n}$ is in weak ${\mathrm{\ell }}_{1}$ but not in ${\mathrm{\ell }}_{1}$ .

For $p$ , ${p}^{\mathrm{\prime }}$ such that ${p}^{\mathrm{\prime }}\mathrm{>}p$ , we have ${\mathrm{\ell }}_{p}\mathrm{\subset }w{\mathrm{\ell }}_{p}\mathrm{\subset }{\mathrm{\ell }}_{{p}^{\mathrm{\prime }}}$ .

Let $\mathcal{D}\mathrm{=}B$ be an orthonormal basis for the Hilbert space $\mathbf{X}\mathrm{=}H$ . For $f\mathrm{\in }\mathbf{X}$ with representation in $B\mathrm{=}\left[{\varphi }_{1},{\varphi }_{2},\mathrm{\dots }\right]$ as $f\mathrm{=}{\mathrm{\sum }}_{n}{c}_{n}\left(f\right){\varphi }_{n}$ , we have $f\mathrm{\in }{\mathrm{A}}^{r}{\left(B\right)}_{\mathbf{X}}$ if and only if the sequence $\left({c}_{n}\left(f\right)\right)\mathrm{\in }w{\mathrm{\ell }}_{\tau }$ , with $\frac{1}{\tau }\mathrm{=}r\mathrm{+}\frac{1}{2}$ . Moreover, there exist ${C}_{0},{C}_{0}^{\mathrm{\prime }}\mathrm{\in }\mathbb{R}$ such that ${C}_{0}^{\mathrm{\prime }}{\parallel \left({c}_{n}\left(f\right)\right)\parallel }_{w{\mathrm{\ell }}_{\tau }}\mathrm{\le }{\parallel f\parallel }_{{\mathrm{A}}^{r}}\mathrm{\le }{C}_{0}{\parallel \left({c}_{n}\left(f\right)\right)\parallel }_{w{\mathrm{\ell }}_{\tau }}$ .

Let $r=\frac{1}{2}$ . $f\in {\mathcal{A}}^{\frac{1}{2}}$ if and only if $\left({c}_{n}\left(f\right)\right)\in w{\mathrm{\ell }}_{\tau }$ , i.e. if ${c}_{n}^{*}\left(f\right)\le M{n}^{-1}=\frac{M}{n}$ .

We prove the converse statement; the forward statement proof is left to the reader. We would like to show thatif $\left({c}_{n}\left(f\right)\right)\in w{\mathrm{\ell }}_{\tau }$ , then $f\in {\mathcal{A}}^{r}$ , with $r=\frac{1}{\tau }-\frac{1}{2}$ . The best $n$ -term approximation of $f$ in $B$ is of the form $s={\sum }_{k\in \mathrm{\Lambda }}{a}_{k}{\varphi }_{k},\mathrm{♯}\mathrm{\Lambda }\le n$ . Therefore, we have:

where $M:=\parallel \left(c{}_{n}\left(f\right)\parallel {}_{w{\mathrm{\ell }}_{p}}$ .

We prove the converse statement; the forward statement proof is left to the reader. We would like to show thatif $\left({c}_{n}\left(f\right)\right)\in w{\mathrm{\ell }}_{\tau }$ , then $f\in {\mathcal{A}}^{r}$ , with $r=\frac{1}{\tau }-\frac{1}{2}$ . The best $n$ -term approximation of $f$ in $B$ is of the form $s={\sum }_{k\in \mathrm{\Lambda }}{a}_{k}{\varphi }_{k},\mathrm{♯}\mathrm{\Lambda }\le n$ . Therefore, we have:

where we define $C=\frac{{\lambda }_{0}}{r}$ . Using this result in the earlier statement, we get $\begin{array}{cc}\text{(5)}& \begin{array}{ccc}\hfill \sum _{k=n+1}^{\mathrm{\infty }}{|{c}_{k}^{*}\left(f\right)|}^{2}& \hfill \le \hfill & C{M}^{2}{n}^{-2r}\le {M}^{2}z{n}^{-r};\hfill \end{array}\end{array}$

this implies by definition that $\left({c}_{k}\left(f\right)\right)\in {\mathcal{A}}^{r}$ .

so some one know about replacing silicon atom with phosphorous in semiconductors device?
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!