en.wikipedia.org website review
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SEO Keyword summary for en.wikipedia.org/wiki/karhunen%e2%80%93lo%c3%a8ve_theorem
Keywords are extracted from the main content of your website and are the primary indicator of the words this page could rank for. By frequenty count we expect your focus keyword to be displaystyle
Focus keyword
Short and long tail
Short Tail Keywords displaystyle right left |
long Tail Keywords (2 words) karhunenlove transform karhunenlove theorem principal component random variables sidebar hide |
long Tail Keywords (3 words) move to sidebar principal component analysis create account log consists of sinusoidal approximation in bases minimizes the total uncorrelated random variables |
en.wikipedia.org On-Page SEO Scan
Descriptive Elements
The <head> element of a en.wikipedia.org/wiki/karhunen%e2%80%93lo%c3%a8ve_theorem page is used to inform the browser and visitors of the page about the general meta information. The head section of the page is where we place the page title, the definition of the HTML version used, the language of in which the page is written. In the head section we can also include JavaScript and CSS (markup) files for the page.
Page title
Title length
kosambindashkarhunenndashloegraveve theorem wikipedia
Meta description
Meta description legth
Meta description SEO
No meta relevance in the description detected !
Content SEO
Number of Words
Spam detected?
Headings
Heading distribution
Heading normalisation
Heading SEO impact
Emphasis (bold and italic)
Emphasis SEO impact
Images
Number of images
Images dimensions
Image alt descriptions
Images SEO impact
wikipedia free encyclopedia displaystyle xtsum infty zkekt forall tin abqquad xtin omega fmathbf quad textie mathbf tsin kxstmathbf xsxt kxst stin beginalignedtkxl abto abfmapsto tkxfint abkxscdot fsdsendaligned int abkxsteksdslambda kekt zkint abxtektdt kin mathbb qquad mboxandqquad zizjdelta ijlambda jforall ijin kxstsum lambda keksekt beginalignedmathbf zkmathbf leftint abxtektdtrightint abmathbf xtektdt ptmathbf zizjmathbf abint abxtxsejteisdtdsrightint leftxtxsrightejteisdtdsint abkxstejteisdtdsint abeisleftint abkxstejtdtrightdslambda jint abeisejsdsdelta jendaligned snsum nzkekt leftleftxtsnright rightmathbf leftxt leftsn right leftxtsnrightkxttmathbf leftsum nsum nzkzell ekteell tright leftxtsum nzkektrightkxttsum nlambda nint abxtxseksektdsrightkxttsum endaligned lim nto sum neitziomega xtomega akomega fkt abxtomega fktdt hat nakomega varepsilon ntsum beginalignedvarepsilon tmathbf aiomega ajomega fitfjtrightsum abxtxsfitfjsdsdtrightfitfjtsum fitfjtint abkxstfitfjsdsdtendaligned abvarepsilon tdtsum abkxstfktfksdsdt erfktkin ldots abkxstfktfksdsdtbeta kleftint abfktfktdt frac partial erpartial fitint ableftint abkxstfisdsbeta ifitrightdt ifit operatorname var ekt zksum aboperatorname xtdtsum ksum nin kgeq wkvarphi varphi pkmathbb xtl hvarphi ipklogpk hek ngeq nzkektxtrightl leq nwkvarphi ktxtrightl ell nmathbb well twkvarphi ktl kxtl xtwkvarphi npkgeq nqk pklogpkleq qklogqk leftgmright mleq ysum leftlangle ygmrightrangle yngmn ymsum mmathbf leftleftyymright rightsum mmn leftleftleftlangle rnmmathbf leftynymright leftleftlangle yxrangle rightlangle kxxrangle rnmxnxm msum kgmgmrightrangle geq kgm rightrangle fin mathrm leftgmrightmin fmsum min imleftlangle fgmrightrangle mleftleftffmright mnotin imn rightrightmin fgmkrightrangle rightgeq fgmk mleftlangle gmk theta tleftleftlangle rightgm fgmmrightrangle rightqquad txbegincasesxxgeq xtendcases fmathrm pleftsum rightprightfrac pinfty moleftm pright qinfty ynfnpbmod prppfrac nqquad ynfrac fnpbmod rnkmathbf ynykfrac nfkpbmod nfrac nftheta bar fnkquad fnfn rnkrynkqquad rykfrac leftfrac sqrt nei mnnright pymhat rymfrac nlefthat fmright fndelta ndelta leftgmndelta nmright mnnrightrangle rightpymfrac nsin knright leftxnrightnin xleftx xnrightt xxmu sigma ijmathbf xixjqquad nsigma ijejlambda eiquad leftrightarrow elambda ent ivarphi iiin beginalignedphi leftvarphi nrighttphi tphi iendaligned xsum nlangle ixrangle isum nvarphi itxvarphi begincasesyphi txxphi yendcases yivarphi itx nyivarphi klambda igeq alpha kwtsoperatorname cov wtwsminst ektsqrt sin leftleftktfrac rightpi kfrac beginalignedint abkwstesdslambda etqquad tleq minstesdslambda tsesdstint esdslambda etlambda etasin tsqrt rightbcos cos kleftfrac ektasin leftkfrac trightqquad tdt implies asqrt wtsqrt zkfrac leftleftkfrac trightleftkfrac btwttw kbtsmintsts btsum sinkpi tkpi icon rntsentns hxtnt kxtntstquad rnttfrac delta snfbegincasesfrac fwendcases tfrac text within xixidelta tdelta sisidelta hxini kxinisii mathcal lunderline xlog sixisi tsum nsixisum nsidelta txidelta tgtrless cdot gint tstxtdt gunderline rightarrow hgsim nleft eright kgsim nleftetfrac erightdgrightarrow ephi beta erightdg phi esqrt erightphi leftsqrt niphi itquad niint ntphi itdt rnts trntsphi isdslambda iphi nilambda stsum siphi siint tstphi xiint txtphi itdtni nisi overline dots fhxt ttfhunderline xprod iexp iright xisi fkxtmid ttfkunderline gsum sixilambda ktsum isiphi tktxtdt trntsksdssum isiint isdssum itst trntsksdsst tsksdsstrightarrow ktcstquad gmid hint tktmathbf xtmid hdt kint kdtint tktstdtequiv rho mid tint tktksrntsdtdsint tktleftint tksrntsdsrightint tktstdtrho hmathbf tktksmathbf xtxsdtdsint tktksrntsstssdtdsrho kmathbf nrho dgphi tktstdtint tst dte rntau ebtau snffrac jfrac ytnt ytntxtquad rxtsextxs xiphi trxtsphi yiint tytphi itdtint tntxtphi itnixi tntphi yini yinixi fymid cesum itfrac llnlambda ifrac widehat xifrac lkfrac yiwidehat ftsum fiphi itgtsum giphi fiint tftphi itdtquad giint tgtphi figiint tgtftdt tytwidehat tqtsysds tqtsrxstdstfrac qtlambda rxtlambda qtshtshstint thlambda thslambda dlambda wikimedia foundation powered mediawiki
Mobile SEO en.wikipedia.org/wiki/karhunen%e2%80%93lo%c3%a8ve_theorem
Mobile rendering
Mobile optimizations
Responsive design detected (mobile css)
No flash detected !
Mobile improvement
Marketing / lead generation for en.wikipedia.org/wiki/karhunen%e2%80%93lo%c3%a8ve_theorem
Social Media
Facebook shares | Facebook likes | ||
Facebook comments | Tweets | ||
Google +1 |
Conversion form
Search form
Analytics
Online presence
SERP Preview
SERP Title
SERP Link
SERP Description
Domain Level SEO
Domain name
16 characters long
Domain name SEO Impact
Path name
karhunen found in path !
theorem found in path !
Structured data
Publisher Markup
Other Structured data
Website configuration
Correct processing of non-existing pages?
Favicon icon found?
Robots.txt found?
Sitemap found?
Navigation and internal links
Navigation
Url seperator
Human readable urls
Number of links
Link SEO Impact
statistics
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en.m.wikipedia.org |
en.wikipedia.org wikimedia foundation inc
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w permanent link
printable version
karhunenlove theorem
adding to it
httpsenwikipediaorgwindexphptitlekosambikarhunenlovetheoremoldid1245970687
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wiki stochastic process
kari karhunen
michel love
orthogonal
fourier series
hotelling
eigenvector
principal component analysis
orthonormal basis
mean squared error
sinusoidal functions
sine
cosine
random variables
covariance function
uncorrelated
gaussian
stochastically independent
wiener process
empirical
proper orthogonal decomposition
empirical orthogonal functions
meteorology
geophysics
probability space
squareintegrable
linear operator
hilbertschmidt integral operator
integral equation
mercers theorem
functional derivative
bienaym formula
entropy
mean vector
brownian motion
brownian bridge
adaptive optics
singular value decomposition
maximum likelihood
neymanpearson optimum detector
wienerhopf equation
additive white gaussian noise awgn
orthogonality principle
polynomial chaos
reproducing kernel hilbert space
bibcode
citeseerx
s2cid
wayback machine
isbn
arxiv
harvey mudd college
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Links to external pages
Outloing links
fa.wikipedia.org
nl.wikipedia.org
pt.wikipedia.org
ru.wikipedia.org
sv.wikipedia.org
uk.wikipedia.org
www.wikidata.org
www.wikidata.org
ui.adsabs.harvard.edu
citeseerx.ist.psu.edu
www.doi.org
api.semanticscholar.org
fourier.eng.hmc.edu
web.archive.org
www.openlibrary.org
www.openlibrary.org
ui.adsabs.harvard.edu
www.doi.org
www.arxiv.org
ui.adsabs.harvard.edu
www.doi.org
api.semanticscholar.org
reference.wolfram.com
foundation.wikimedia.org
foundation.wikimedia.org
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SEO Advice for en.wikipedia.org
In this section we provide pointers on how you can to optimize your web page so it can be found more easily by search engines and how to make it rank higher by optimizing the content of the page itself. For each of the individual criteria the maximum score is 100%. A score below 70% is considered to be indication that the page is not complying with general SEO standards and should be evaluated and/or fixed. Not every factor is weighted the same and some are not as important as others. Relatively unimportant factors like meta keywords are not included in the overall score.
Item | Factor | Pointers | |
---|---|---|---|
PageTitle | 100% | Far too many sites lack a page title. A page title is the first thing that shows in the search results so always use the title element. | |
Title relevance | 43% | A title should reflect the contents of a site. This site has a 33 % match | |
Title Length | 100% | Limit your title to anywhere between 40 and 70 characters. Your title was 62 characters long | |
Meta Description | 0% | A meta description is the second element that shows in the search results so always use the meta description. | |
Meta description length | 0% | The meta description should be between 145 and 160 characters. This meta description is 1 characters long. | |
Meta description relevance | 0% | Meta Description should reflect the contents of a site. This site has a 0 % match | |
Number of internal links | 85% | Linking to internal pages makes pages easier to find for search engines. Try to keep the number of links on your page roughly below 100. There are 122 internal links on this page. | |
Folder structure | 100% | We found a folder structure in the links on your page. A good folder structure makes a site easier to navigate. We found 3 level 1 folders and 6 folders above or in the first level of navigation. | |
Headings | 37% | Headers should reflect the contents of a site. This site has a 16 % match | |
Links | 12% | Link anchors should to some degree reflect the contents of a site. This site has a 6 % match | |
Image alt tags | 8% | Image alt tags should to some degree reflect the contents of a site. This site has a 3 % match | |
Bold and italic | 24% | Bold and italic tags should reflect the contents of a site to some degree. This site has a 8 % match | |
Html ratio | 30% | Try to keep the html / text ratio as low as possible. More html means longer loading times. Layout should be handled in a serpate css file | |
Image descriptions | 99% | 99.141630901288 % of all images have been described via the "alt" attribute. Describing images with relevant text may lead to better results in the search engines. | |
Page errors | 100% | Pages with no errors display significantly faster on most browsers. We detected 0 errors and warnings | |
WordCount | 20% | An ideal page contains between 400 and 600 words.This page contains 9474 words | |
Server response time | 30% | A slow server slows down a website. This server responds 283.82% slower the average | |
Gzip compression | 30% | This site does not use Gzip compression. Pages may not display as fast as they could | |
Keywords in Domainname | 30% | There are no important keywords in your domain name | |
Keywords in domain path | 100% | There are important keywords in the domain path | |
Structured Data | 100% | Structured data makes it easier for search engines to index your website | |
Inline css | 0% | Do not use inline css declarations. Inline css will slow down the rendering of the website. We detected 486 inline style declarations ( <a style="color:green">) with a size of 16737 bytes | |
Excessive use of the same words | 100% | There is no indication that there are one or more keywords that are used excessively. | |
Frames or iframes | 100% | Perfect, detected not (i)frames on your webpagina | |
Flash | 100% | Perfect, we detected no flash objects on your page | |
Css | 30% | We detected too much (2) CSS files on your page. Css files block the loading of a webpage. | |
Javascript | 100% | Perfect, we did not detect too many blocking JavaScript files | |
Mobile Website | 100% | Perfect, we found a responsive design for mobile users | |
Most important heading | 100% | Perfect, we detected a correct use of the most important (h1) heading! | |
Normalized headings | 40% | We dit not font a normalized heading structure. A heading 2 (h2) for example should be followed by a heading of an equal level (h2), a child heading (h3) or even a aprent heading (h1). |
How would you like to have SEO advice for all your pages ?? Start your SEO Dashboard and optimize your website!
en.wikipedia.org images and descriptions
220 images found at en.wikipedia.org Images can improve the user experience for a website by making a pag visually appealing Images can also add extra keyword relevance to a webpage by using alt tags. Images can also slow down a website. If the width and height for a picture is not specified for a browser know in advance how large the image is. A browser must first load the picture and see before it knows how much space should be on the page. Upon reservation In the meantime, the browser can do little but wait. When the height and width for the plate are given in the HTML code, a browser just continues to build for a page while the images load in the background.
http://en.wikipedia.org/static/images/icons/wikipedia.png height: 50 width: 50 description: no alt description found |
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http://en.wikipedia.org/static/images/mobile/copyright/wikipedia-wordmark-en.svg height: height attribute not set width: width attribute not set description: wikipedia |
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http://en.wikipedia.org/static/images/mobile/copyright/wikipedia-tagline-en.svg height: 13 width: 117 description: the free encyclopedia |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ce1868de371ca24e8d3d14ad49dc03b57990eaf1 height: height attribute not set width: width attribute not set description: {\displaystyle k_{x}(s,t)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/48065dfd97d7b22fcfd2b3b1e4982485989d5d60 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}&t_{k_{x}}&:l^{2}([a,b])&\to l^{2}([a,b])\\&&:f\mapsto t_{k_{x}}f&=\int _{a}^{b}k_{x}(s,\cdot )f(s)\,ds\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/23223c32fd5054a22a855f70feffce7a9c48f868 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {e} [z_{k}]=0,~\forall k\in \mathbb {n} \qquad {\mbox{and}}\qquad \mathbf {e} [z_{i}z_{j}]=\delta _{ij}\lambda _{j},~\forall i,j\in \mathbb {n} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4ce33680637c8657a8bd6a180225c18f165917c3 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\mathbf {e} [z_{k}]&=\mathbf {e} \left[\int _{a}^{b}x_{t}e_{k}(t)\,dt\right]=\int _{a}^{b}\mathbf {e} [x_{t}]e_{k}(t)dt=0\\[8pt]\mathbf {e} [z_{i}z_{j}]&=\mathbf {e} \left[\int _{a}^{b}\int _{a}^{b}x_{t}x_{s}e_{j}(t)e_{i}(s)\,dt\,ds\right]\\&=\int _{a}^{b}\int _{a}^{b}\mathbf {e} \left[x_{t}x_{s}\right]e_{j}(t)e_{i}(s)\,dt\,ds\\&=\int _{a}^{b}\int _{a}^{b}k_{x}(s,t)e_{j}(t)e_{i}(s)\,dt\,ds\\&=\int _{a}^{b}e_{i}(s)\left(\int _{a}^{b}k_{x}(s,t)e_{j}(t)\,dt\right)\,ds\\&=\lambda _{j}\int _{a}^{b}e_{i}(s)e_{j}(s)\,ds\\&=\delta _{ij}\lambda _{j}\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a8931f37eae91505769b097b93769ecd5f5894f4 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\mathbf {e} \left[\left|x_{t}-s_{n}\right|^{2}\right]&=\mathbf {e} \left[x_{t}^{2}\right]+\mathbf {e} \left[s_{n}^{2}\right]-2\mathbf {e} \left[x_{t}s_{n}\right]\\&=k_{x}(t,t)+\mathbf {e} \left[\sum _{k=1}^{n}\sum _{l=1}^{n}z_{k}z_{\ell }e_{k}(t)e_{\ell }(t)\right]-2\mathbf {e} \left[x_{t}\sum _{k=1}^{n}z_{k}e_{k}(t)\right]\\&=k_{x}(t,t)+\sum _{k=1}^{n}\lambda _{k}e_{k}(t)^{2}-2\mathbf {e} \left[\sum _{k=1}^{n}\int _{a}^{b}x_{t}x_{s}e_{k}(s)e_{k}(t)\,ds\right]\\&=k_{x}(t,t)-\sum _{k=1}^{n}\lambda _{k}e_{k}(t)^{2}\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ceffe92a52436b7040a095a973c06a5db4a97624 height: height attribute not set width: width attribute not set description: {\displaystyle \varepsilon _{n}(t)=\sum _{k=n+1}^{\infty }a_{k}(\omega )f_{k}(t)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/db3545962d3e0d868cffefe4b7b028aa583d140a height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\varepsilon _{n}^{2}(t)&=\mathbf {e} \left[\sum _{i=n+1}^{\infty }\sum _{j=n+1}^{\infty }a_{i}(\omega )a_{j}(\omega )f_{i}(t)f_{j}(t)\right]\\&=\sum _{i=n+1}^{\infty }\sum _{j=n+1}^{\infty }\mathbf {e} \left[\int _{a}^{b}\int _{a}^{b}x_{t}x_{s}f_{i}(t)f_{j}(s)\,ds\,dt\right]f_{i}(t)f_{j}(t)\\&=\sum _{i=n+1}^{\infty }\sum _{j=n+1}^{\infty }f_{i}(t)f_{j}(t)\int _{a}^{b}\int _{a}^{b}k_{x}(s,t)f_{i}(t)f_{j}(s)\,ds\,dt\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a21063ed7c70d9c06aa9e8ea756049df74acd26d height: height attribute not set width: width attribute not set description: {\displaystyle er[f_{k}(t),k\in \{n+1,\ldots \}]=\sum _{k=n+1}^{\infty }\int _{a}^{b}\int _{a}^{b}k_{x}(s,t)f_{k}(t)f_{k}(s)\,ds\,dt-\beta _{k}\left(\int _{a}^{b}f_{k}(t)f_{k}(t)\,dt-1\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a034ea2ca9cdff033c1bc016f615dd223cf6ffeb height: height attribute not set width: width attribute not set description: {\displaystyle x_{t}=\sum _{k=1}^{\infty }w_{k}\varphi _{k}(t)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/56131afa8525a4f8306020af0a6123b76d80f627 height: height attribute not set width: width attribute not set description: {\displaystyle \varphi _{k}(t)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/aea7ab451421a62d7de329faed9aa86419bebeab height: height attribute not set width: width attribute not set description: {\displaystyle w_{k}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d508243f2e679ea109f53d5f05dab97da287d79d height: height attribute not set width: width attribute not set description: {\displaystyle p_{k}=\mathbb {e} [|w_{k}|^{2}]/\mathbb {e} [|x_{t}|_{l^{2}}^{2}]} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/12e9bffadf6daf9ffc45c59bcc9faebb1ffcf664 height: height attribute not set width: width attribute not set description: {\displaystyle \sum _{k=1}^{\infty }p_{k}=1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8e605284f75f7dd8e38bf4faf01fa647745f650f height: height attribute not set width: width attribute not set description: {\displaystyle h(\{\varphi _{k}\})=-\sum _{i}p_{k}\log(p_{k})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a00c0ca838b9c6466607f731035415c774b60560 height: height attribute not set width: width attribute not set description: {\displaystyle h(\{\varphi _{k}\})\geq h(\{e_{k}\})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/dad86b92f0c76d343e12c4a90b368834329bd5d6 height: height attribute not set width: width attribute not set description: {\displaystyle \varphi _{k}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/740342d6aa43a63a2c81d267329cff4f4e508f42 height: height attribute not set width: width attribute not set description: {\displaystyle e_{k}(t)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/01084a31964201514f3e6bd0136989e11ea6e58a height: height attribute not set width: width attribute not set description: {\displaystyle p_{k}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f27215e46abcad60f100434d2c8003310580af95 height: height attribute not set width: width attribute not set description: {\displaystyle q_{k}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d5ea1c340889c82b8f87bf48be5409eb22872947 height: height attribute not set width: width attribute not set description: {\displaystyle n\geq 1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1a24f4c64db82ae2f1e89d9861304aec8598c7e5 height: height attribute not set width: width attribute not set description: {\displaystyle e_{k}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c6d172e7410152d0942df1c2ce7a406be91ec189 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {e} \left|\sum _{k=1}^{n}z_{k}e_{k}(t)-x_{t}\right|_{l^{2}}^{2}\leq \mathbb {e} \left|\sum _{k=1}^{n}w_{k}\varphi _{k}(t)-x_{t}\right|_{l^{2}}^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0c9dd795699c6f791be251af56fda845082a2cb9 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {e} \left|\sum _{k=1}^{n}w_{k}\varphi _{k}(t)-x_{t}\right|_{l^{2}}^{2}=\mathbb {e} |x_{t}^{2}|_{l^{2}}+\sum _{k=1}^{n}\sum _{\ell =1}^{n}\mathbb {e} [w_{\ell }\varphi _{\ell }(t)w_{k}^{*}\varphi _{k}^{*}(t)]_{l^{2}}-\sum _{k=1}^{n}\mathbb {e} [w_{k}\varphi _{k}x_{t}^{*}]_{l^{2}}-\sum _{k=1}^{n}\mathbb {e} [x_{t}w_{k}^{*}\varphi _{k}^{*}(t)]_{l^{2}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/82120d04dfb3cbadc4912951dd12b5568c9cd8f3 height: height attribute not set width: width attribute not set description: {\displaystyle x_{t}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bd4cc80ff88af90eeed172031bedda8d116e2298 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {e} [x_{t}]_{l^{2}}^{2}-\sum _{k=1}^{n}\mathbb {e} [|w_{k}|^{2}]} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d7b9f649b35e5c0ccd0d9294dddd5ed2c38df531 height: height attribute not set width: width attribute not set description: {\displaystyle {\displaystyle \mathbb {e} [x_{t}]_{l^{2}}^{2}-\sum _{k=1}^{n}\mathbb {e} [|z_{k}|^{2}]}\leq {\displaystyle \mathbb {e} [x_{t}]_{l^{2}}^{2}-\sum _{k=1}^{n}\mathbb {e} [|w_{k}|^{2}]}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5f66e132a304afedacd6c2503f0484633d16ee81 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {e} [|x_{t}|_{l^{2}}^{2}]} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5b0b4403d81d04c1e0496e15d3188760d6eaae0f height: height attribute not set width: width attribute not set description: {\displaystyle \sum _{k=1}^{n}p_{k}\geq \sum _{k=1}^{n}q_{k}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c94fe9e57e1237aa3a6968d2ba36c6d91a9cd749 height: height attribute not set width: width attribute not set description: {\displaystyle -\sum _{k=1}^{\infty }p_{k}\log(p_{k})\leq -\sum _{k=1}^{\infty }q_{k}\log(q_{k})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bcd3f4702e010346edfff5ba80842c3de529661d height: height attribute not set width: width attribute not set description: {\displaystyle \left\{g_{m}\right\}_{0\leq m\leq n}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/47e9e906fe459b0d3818e61ef1d23a002cd910dd height: height attribute not set width: width attribute not set description: {\displaystyle y=\sum _{m=0}^{n-1}\left\langle y,g_{m}\right\rangle g_{m},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/02f1fe093bbcd1aa4a590ba40632868931607175 height: height attribute not set width: width attribute not set description: {\displaystyle \left\langle y,g_{m}\right\rangle =\sum _{n=0}^{n-1}{y[n]}g_{m}^{*}[n]} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c01425e32ce9f90d70c69f766b7e9e150dacc4 height: height attribute not set width: width attribute not set description: {\displaystyle y_{m}=\sum _{m=0}^{m-1}\left\langle y,g_{m}\right\rangle g_{m}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f03ea5618ab818ac1fba489f606629de7bae0624 height: height attribute not set width: width attribute not set description: {\displaystyle \varepsilon [m]=\mathbf {e} \left\{\left\|y-y_{m}\right\|^{2}\right\}=\sum _{m=m}^{n-1}\mathbf {e} \left\{\left|\left\langle y,g_{m}\right\rangle \right|^{2}\right\}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/97040a6e6c6093e9b2025f61eef9d5b4c01e96a7 height: height attribute not set width: width attribute not set description: {\displaystyle r[n,m]=\mathbf {e} \left\{y[n]y^{*}[m]\right\}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/329f1d64474e88691438cbf535cef5909c6afe90 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {e} \left\{\left|\langle y,x\rangle \right|^{2}\right\}=\langle kx,x\rangle =\sum _{n=0}^{n-1}\sum _{m=0}^{n-1}r[n,m]x[n]x^{*}[m]} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/495b9ad300b894a7656069ceedb341799830137d height: height attribute not set width: width attribute not set description: {\displaystyle \varepsilon [m]=\sum _{m=m}^{n-1}{\left\langle kg_{m},g_{m}\right\rangle }} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ada8bd92150dcbf57702d1f335b74be240e84a28 height: height attribute not set width: width attribute not set description: {\displaystyle \varepsilon [m]=\sum _{m=m}^{n-1}\left\langle kg_{m},g_{m}\right\rangle } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/06b5a4294150613b5ba8de7102524f000498fdf6 height: height attribute not set width: width attribute not set description: {\displaystyle \left\{g_{m}\right\}_{0\leq m |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/897541c95995cf4fa433bf0a7daf6a547f89a220 height: height attribute not set width: width attribute not set description: {\displaystyle \left\langle kg_{m},g_{m}\right\rangle \geq \left\langle kg_{m+1},g_{m+1}\right\rangle ,\qquad 0\leq m |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fe05dcfa4489e2d710415575d723411d19da27d5 height: height attribute not set width: width attribute not set description: {\displaystyle f\in \mathrm {h} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/32db8e791eaa12e32afc8fc1d60386643e43e315 height: height attribute not set width: width attribute not set description: {\displaystyle \mathrm {h} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/393b2965cb390bb885d4f0247371cf4ca33872d7 height: height attribute not set width: width attribute not set description: {\displaystyle \mathrm {b} =\left\{g_{m}\right\}_{m\in \mathbb {n} }} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b53fcb87b0f4d69be22cc6a9d2fb227dd2da91d2 height: height attribute not set width: width attribute not set description: {\displaystyle f_{m}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fdc0574c02e3a3a24f73420ca22a6473c0e04af6 height: height attribute not set width: width attribute not set description: {\displaystyle f_{m}=\sum _{m\in i_{m}}\left\langle f,g_{m}\right\rangle g_{m}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6a2c7e02bbbe9afe5d05b3e7a225d12d294cc5fb height: height attribute not set width: width attribute not set description: {\displaystyle \varepsilon [m]=\left\{\left\|f-f_{m}\right\|^{2}\right\}=\sum _{m\notin i_{m}}^{n-1}\left\{\left|\left\langle f,g_{m}\right\rangle \right|^{2}\right\}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8de31aa8e674740480b1b77e7b32c1ae28bbe03b height: height attribute not set width: width attribute not set description: {\displaystyle \left|\left\langle f,g_{m}\right\rangle \right|.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e9373e827e06f9dabed3e182822845d759959d97 height: height attribute not set width: width attribute not set description: {\displaystyle \left\{\left|\left\langle f,g_{m}\right\rangle \right|\right\}_{m\in \mathbb {n} }} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/41a3cfc0090aff9de1459dafb8f39d59c26a7492 height: height attribute not set width: width attribute not set description: {\displaystyle \left|\left\langle f,g_{m_{k}}\right\rangle \right|\geq \left|\left\langle f,g_{m_{k+1}}\right\rangle \right|.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cf006c4ae061880324b91a4899c7a0e611a5e138 height: height attribute not set width: width attribute not set description: {\displaystyle f_{m}=\sum _{k=1}^{m}\left\langle f,g_{m_{k}}\right\rangle g_{m_{k}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1777b566936fff422629023f5dd1373eeb262425 height: height attribute not set width: width attribute not set description: {\displaystyle f_{m}=\sum _{m=0}^{\infty }\theta _{t}\left(\left\langle f,g_{m}\right\rangle \right)g_{m}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/22e901c39f2ac9b9a0864392401745fb7e0c98a2 height: height attribute not set width: width attribute not set description: {\displaystyle t=\left|\left\langle f,g_{m_{m}}\right\rangle \right|,\qquad \theta _{t}(x)={\begin{cases}x&|x|\geq t\\0&|x| |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a10028719c0a213e358f42be10ff625fcd480c61 height: height attribute not set width: width attribute not set description: {\displaystyle \varepsilon [m]=\left\{\left\|f-f_{m}\right\|^{2}\right\}=\sum _{k=m+1}^{\infty }\left\{\left|\left\langle f,g_{m_{k}}\right\rangle \right|^{2}\right\}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/48d2786da17e091c0b52ca55bd0a31d6aaef6f71 height: height attribute not set width: width attribute not set description: {\displaystyle \left|\left\langle f,g_{m_{k}}\right\rangle \right|} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cef3b526868f71c052da5a93c7f8f95a6242beeb height: height attribute not set width: width attribute not set description: {\displaystyle \mathrm {i} ^{\mathrm {p} }} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fe0fefd05963688c8ea2748e96309467b20ab08f height: height attribute not set width: width attribute not set description: {\displaystyle \|f\|_{\mathrm {b} ,p}=\left(\sum _{m=0}^{\infty }\left|\left\langle f,g_{m}\right\rangle \right|^{p}\right)^{\frac {1}{p}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7f5c71fe6db783fe6fff582bbb5db8e1b9baa10d height: height attribute not set width: width attribute not set description: {\displaystyle \|f\|_{\mathrm {b} ,p}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/772e21fdc7ea1d67858484416b50503f75496e35 height: height attribute not set width: width attribute not set description: {\displaystyle \|f\|_{\mathrm {b} ,p}<\infty } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1c5514c4d3d4296125778f3546ddee2b05272abf height: height attribute not set width: width attribute not set description: {\displaystyle \varepsilon [m]\leq {\frac {\|f\|_{\mathrm {b} ,p}^{2}}{{\frac {2}{p}}-1}}m^{1-{\frac {2}{p}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/626974ced6297faf4103543175e6afe5a9ee7236 height: height attribute not set width: width attribute not set description: {\displaystyle \varepsilon [m]=o\left(m^{1-{\frac {2}{p}}}\right).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/97c2c3d7907e4a58e3224371cb6647948350d1c1 height: height attribute not set width: width attribute not set description: {\displaystyle \varepsilon [m]=o\left(m^{1-{\frac {2}{p}}}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/34e0c0a673d59203daadc33b9985aaf65bef3af3 height: height attribute not set width: width attribute not set description: {\displaystyle \|f\|_{\mathrm {b} ,q}<\infty } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8491175ed9cde718f570ca5a1690e598acc83818 height: height attribute not set width: width attribute not set description: {\displaystyle \sum _{n=0}^{n-1}f[n]=0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/719334ff76852253bde14fe3e5ea7d76df8937dc height: height attribute not set width: width attribute not set description: {\displaystyle y[n]=f[(n-p){\bmod {n}}]} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4714f33eaf2249c07c414354ca7095725ef0edfc height: height attribute not set width: width attribute not set description: {\displaystyle \pr(p=p)={\frac {1}{n}},\qquad 0\leq p |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2fcbf5841519051f55abe7088d6220ea4dd73705 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {e} \{y[n]\}={\frac {1}{n}}\sum _{p=0}^{n-1}f[(n-p){\bmod {n}}]=0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6e40a9d9083e7fd94b363d97fed5f876131e1891 height: height attribute not set width: width attribute not set description: {\displaystyle r[n,k]=\mathbf {e} \{y[n]y[k]\}={\frac {1}{n}}\sum _{p=0}^{n-1}f[(n-p){\bmod {n}}]f[(k-p){\bmod {n}}]={\frac {1}{n}}f\theta {\bar {f}}[n-k],\quad {\bar {f}}[n]=f[-n]} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5a8cfb5801399b3718406c3c48962f205026f127 height: height attribute not set width: width attribute not set description: {\displaystyle r[n,k]=r_{y}[n-k],\qquad r_{y}[k]={\frac {1}{n}}f\theta {\bar {f}}[k]} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d5a07628c1bf3ecce4b04d1876ee7f15c6b7abf1 height: height attribute not set width: width attribute not set description: {\displaystyle \left\{{\frac {1}{\sqrt {n}}}e^{i2\pi mn/n}\right\}_{0\leq m |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ae2ed523a89b792ecc1f0024675c39466bd7c8ae height: height attribute not set width: width attribute not set description: {\displaystyle p_{y}[m]={\hat {r}}_{y}[m]={\frac {1}{n}}\left|{\hat {f}}[m]\right|^{2}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/24ed776f976f118ac0a033bb5cdaaaf0ebb97edb height: height attribute not set width: width attribute not set description: {\displaystyle f[n]=\delta [n]-\delta [n-1]} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/007c54c9fb2f295a49f42934713f2249674b54dd height: height attribute not set width: width attribute not set description: {\displaystyle \left\{g_{m}[n]=\delta [n-m]\right\}_{0\leq m |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4d4950a653a45d50571c71ad9019ce11467431e0 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {e} \left\{\left|\left\langle y[n],{\frac {1}{\sqrt {n}}}e^{i2\pi mn/n}\right\rangle \right|^{2}\right\}=p_{y}[m]={\frac {4}{n}}\sin ^{2}\left({\frac {\pi k}{n}}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c098f6386284a43c68f4120ecaeba5fe583e3d6d height: height attribute not set width: width attribute not set description: {\displaystyle \int _{a}^{b}k_{x}(s,t)e_{k}(s)\,ds=\lambda _{k}e_{k}(t).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7cc7d63f7f5cba027f9ed7f540d94d7c5296ffae height: height attribute not set width: width attribute not set description: {\displaystyle \left(x_{n}\right)_{n\in \{1,\ldots ,n\}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d46e8597f288f446e1fbf98aeb6a27154fdae7 height: height attribute not set width: width attribute not set description: {\displaystyle x=\left(x_{1}~x_{2}~\ldots ~x_{n}\right)^{t}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e63c9374e288a0f01a3f4583454c5087e5ef64e0 height: height attribute not set width: width attribute not set description: {\displaystyle x:=x-\mu _{x}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2bfe6d3f115b8d6cb595119ea9bc7962a11db65a height: height attribute not set width: width attribute not set description: {\displaystyle \mu _{x}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0aca0bb8748f361f4f3051b36cb1b667b357891e height: height attribute not set width: width attribute not set description: {\displaystyle \sigma _{ij}=\mathbf {e} [x_{i}x_{j}],\qquad \forall i,j\in \{1,\ldots ,n\}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2c57f3d10919f5f702e973d300f94f5312a20531 height: height attribute not set width: width attribute not set description: {\displaystyle \sum _{j=1}^{n}\sigma _{ij}e_{j}=\lambda e_{i}\quad \leftrightarrow \quad \sigma e=\lambda e} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/567f4c28d0daa92adc5ae7098f40bcf650f36c4b height: height attribute not set width: width attribute not set description: {\displaystyle e=(e_{1}~e_{2}~\ldots ~e_{n})^{t}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/12d5be9beb2f7a56cdee3c6563c9453a913a0c92 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {r} ^{n}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2c7d52ef81a413ab076d2ad3c7576cca5d935b61 height: height attribute not set width: width attribute not set description: {\displaystyle \{\lambda _{i},\varphi _{i}\}_{i\in \{1,\ldots ,n\}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4d1e4278d7acfdf88789c5fbedeabacb7a46e1c7 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\phi &:=\left(\varphi _{1}~\varphi _{2}~\ldots ~\varphi _{n}\right)^{t}\\\phi ^{t}\phi &=i\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3613e70626b199b3df4c1784086d518bdf9a0fec height: height attribute not set width: width attribute not set description: {\displaystyle x=\sum _{i=1}^{n}\langle \varphi _{i},x\rangle \varphi _{i}=\sum _{i=1}^{n}\varphi _{i}^{t}x\varphi _{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3c367e0b265bd2c606d6a6ac30aa9aa3c2e84620 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{cases}y=\phi ^{t}x\\x=\phi y\end{cases}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/92e1807191da40b96a2f271b01fad9268eb5818e height: height attribute not set width: width attribute not set description: {\displaystyle y_{i}=\varphi _{i}^{t}x} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/70503774fb21be77396899900d3aa1e47d8f9e10 height: height attribute not set width: width attribute not set description: {\displaystyle \varphi _{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a856e0d4b07113c1dec633569682033024496ed0 height: height attribute not set width: width attribute not set description: {\displaystyle x=\sum _{i=1}^{n}y_{i}\varphi _{i}=\sum _{i=1}^{n}\langle \varphi _{i},x\rangle \varphi _{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/47b70aaa6c3962eb178f37a462bf188fe447cdf3 height: height attribute not set width: width attribute not set description: {\displaystyle k\in \{1,\ldots ,n\}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/dc8e8c17230edea05291d5dfb8443edb6eadff61 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {\sum _{i=1}^{k}\lambda _{i}}{\sum _{i=1}^{n}\lambda _{i}}}\geq \alpha } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/44e5a6da21737f59f86cac50b7c546c2553a2e43 height: height attribute not set width: width attribute not set description: {\displaystyle k_{w}(t,s)=\operatorname {cov} (w_{t},w_{s})=\min(s,t).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/879f6833cb32b27a7b2afc3caf2226b2c04a01c8 height: height attribute not set width: width attribute not set description: {\displaystyle e_{k}(t)={\sqrt {2}}\sin \left(\left(k-{\tfrac {1}{2}}\right)\pi t\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1c41fc0b4b98557a188fcb914277720664e2271c height: height attribute not set width: width attribute not set description: {\displaystyle \lambda _{k}={\frac {1}{(k-{\frac {1}{2}})^{2}\pi ^{2}}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cfc698624c44e61e07c6ec95d7779ffa21e66779 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\int _{a}^{b}k_{w}(s,t)e(s)\,ds&=\lambda e(t)\qquad \forall t,0\leq t\leq 1\\\int _{0}^{1}\min(s,t)e(s)\,ds&=\lambda e(t)\qquad \forall t,0\leq t\leq 1\\\int _{0}^{t}se(s)\,ds+t\int _{t}^{1}e(s)\,ds&=\lambda e(t)\qquad \forall t,0\leq t\leq 1\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/53b878d265ee346bd3a0a88c012aea3249079bdd height: height attribute not set width: width attribute not set description: {\displaystyle \int _{t}^{1}e(s)\,ds=\lambda e'(t)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c75462094ae29ed5eb3b8d78b7beeacc5e573d height: height attribute not set width: width attribute not set description: {\displaystyle -e(t)=\lambda e''(t)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ad1c1ba531c50f92dcb463f37734fc5ac73617fc height: height attribute not set width: width attribute not set description: {\displaystyle e(t)=a\sin \left({\frac {t}{\sqrt {\lambda }}}\right)+b\cos \left({\frac {t}{\sqrt {\lambda }}}\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5e5d46cb78fe185d61f8a4f8f3c94ef8e332f96b height: height attribute not set width: width attribute not set description: {\displaystyle \cos \left({\frac {1}{\sqrt {\lambda }}}\right)=0} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8a112cd10c7d9b6a325b7f9c23563e617ca2d6ca height: height attribute not set width: width attribute not set description: {\displaystyle \lambda _{k}=\left({\frac {1}{(k-{\frac {1}{2}})\pi }}\right)^{2},\qquad k\geq 1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/da5947037a4d045455a75ff5a6ba242a4d919081 height: height attribute not set width: width attribute not set description: {\displaystyle e_{k}(t)=a\sin \left((k-{\frac {1}{2}})\pi t\right),\qquad k\geq 1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3022d53bf23eb69c0c7b1e9cbcd53ee67a115462 height: height attribute not set width: width attribute not set description: {\displaystyle \int _{0}^{1}e_{k}^{2}(t)\,dt=1\quad \implies \quad a={\sqrt {2}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b4ec85c5cbcb72a101751d96c7e4afd0e7aa56d7 height: height attribute not set width: width attribute not set description: {\displaystyle w_{t}={\sqrt {2}}\sum _{k=1}^{\infty }z_{k}{\frac {\sin \left(\left(k-{\frac {1}{2}}\right)\pi t\right)}{\left(k-{\frac {1}{2}}\right)\pi }}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bafa089d991504bb539141c6221e17f79d06d7b8 height: height attribute not set width: width attribute not set description: {\displaystyle t\in [0,1].} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/af3e3e9fa1cbd3c579a27e8a012c617eb0e78335 height: height attribute not set width: width attribute not set description: {\displaystyle b_{t}=w_{t}-tw_{1}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5a4e12e36419548dbbff124bef1def30e402bca3 height: height attribute not set width: width attribute not set description: {\displaystyle k_{b}(t,s)=\min(t,s)-ts} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4b20478f23df011f4dd269d2898fbaeb62226996 height: height attribute not set width: width attribute not set description: {\displaystyle b_{t}=\sum _{k=1}^{\infty }z_{k}{\frac {{\sqrt {2}}\sin(k\pi t)}{k\pi }}} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/1/1c/wiki_letter_w_cropped.svg/20px-wiki_letter_w_cropped.svg.png height: 14 width: 20 description: [icon] |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1f09f8b659fd5e362914b33effdda43332d909e4 height: height attribute not set width: width attribute not set description: {\displaystyle r_{n}(t,s)=e[n(t)n(s)]} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8535c424c877ac343d6a41af305b8185deb88ce7 height: height attribute not set width: width attribute not set description: {\displaystyle h:x(t)=n(t),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b4a7dbda3a256b498a5af869973deda695d2fd17 height: height attribute not set width: width attribute not set description: {\displaystyle k:x(t)=n(t)+s(t),\quad t\in (0,t)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d29d32a2e6e254a533cf3dd58891f0932d9423a6 height: height attribute not set width: width attribute not set description: {\displaystyle r_{n}(t)={\tfrac {1}{2}}n_{0}\delta (t),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b7ea030a0ab547e23e58dbbc7355ecf7eb447e91 height: height attribute not set width: width attribute not set description: {\displaystyle s_{n}(f)={\begin{cases}{\frac {n_{0}}{2}}&|f| |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1bb85d446ed4d9d91672d429942108e541557f47 height: height attribute not set width: width attribute not set description: {\displaystyle {\frac {n}{2\omega }},n\in \mathbf {z} .} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c45874403782b8d0062e4c77cfe40ba9ce87b8ca height: height attribute not set width: width attribute not set description: {\displaystyle \delta t={\frac {n}{2\omega }}{\text{ within }}(0,''t'').} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/645925f6073f6e333645d7b401a0875596c44b41 height: height attribute not set width: width attribute not set description: {\displaystyle x_{i}=x(i\,\delta t)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5d35f2c71e0ddc5302454a3b5db3cbf5fd659f38 height: height attribute not set width: width attribute not set description: {\displaystyle n={\frac {t}{\delta t}}=t(2\omega )=2\omega t} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c09f5f997711f820ce55620d961f7ecce1797be2 height: height attribute not set width: width attribute not set description: {\displaystyle \{x_{1},x_{2},\ldots ,x_{n}\}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d58ee2068cec2a4f6ea35b2e34f81cc183bce4b7 height: height attribute not set width: width attribute not set description: {\displaystyle s_{i}=s(i\,\delta t)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/289170b478319c5c7388eb5c1db0f7272311ea1a height: height attribute not set width: width attribute not set description: {\displaystyle h:x_{i}=n_{i},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b79eb909c0a43021db7455d0ad001eb94fe33e0a height: height attribute not set width: width attribute not set description: {\displaystyle k:x_{i}=n_{i}+s_{i},i=1,2,\ldots ,n.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2aa6c623365498ae585e80aabd9887e0074cb701 height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {l}}({\underline {x}})=\log {\frac {\sum _{i=1}^{n}(2s_{i}x_{i}-s_{i}^{2})}{2\sigma ^{2}}}\leftrightarrow \delta t\sum _{i=1}^{n}s_{i}x_{i}=\sum _{i=1}^{n}s(i\,\delta t)x(i\,\delta t)\,\delta t\gtrless \lambda _{\cdot }2} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9930336d4c95c7926fb5f5aa87ab2c7c9c3934d0 height: height attribute not set width: width attribute not set description: {\displaystyle g=\int _{0}^{t}s(t)x(t)\,dt.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f1845bd4889d58a485a0c0e766057cac63840491 height: height attribute not set width: width attribute not set description: {\displaystyle g({\underline {x}})>g_{0}\rightarrow k |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7e5ea385579b6e2b67b0212a4beb7d75c4386fcf height: height attribute not set width: width attribute not set description: {\displaystyle h:g\sim n\left(0,{\tfrac {1}{2}}n_{0}e\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a1017c2c039c3b9915eab5c3e3920d983003f7e7 height: height attribute not set width: width attribute not set description: {\displaystyle k:g\sim n\left(e,{\tfrac {1}{2}}n_{0}e\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/83c78eef26b181ad61e76b91e85d3796fa255e2d height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {e} =\int _{0}^{t}s^{2}(t)\,dt} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/755bbb34c6ec534d5dbd0c3cee6bae4fe261df96 height: height attribute not set width: width attribute not set description: {\displaystyle \alpha =\int _{g_{0}}^{\infty }n\left(0,{\tfrac {1}{2}}n_{0}e\right)\,dg\rightarrow g_{0}={\sqrt {{\tfrac {1}{2}}n_{0}e}}\phi ^{-1}(1-\alpha )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/87a2c1346fa308cbca018cf067e3f9e193072f6c height: height attribute not set width: width attribute not set description: {\displaystyle \beta =\int _{g_{0}}^{\infty }n\left(e,{\tfrac {1}{2}}n_{0}e\right)\,dg=1-\phi \left({\frac {g_{0}-e}{\sqrt {{\tfrac {1}{2}}n_{0}e}}}\right)=\phi \left({\sqrt {\frac {2e}{n_{0}}}}-\phi ^{-1}(1-\alpha )\right),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4e0b2810a9a6fef382bec8d835be29090f3d6871 height: height attribute not set width: width attribute not set description: {\displaystyle r_{n}(t,s)=e[n(t)n(s)],} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4790260bdcdfce5191472820fe8a8f7015c0b45d height: height attribute not set width: width attribute not set description: {\displaystyle n(t)=\sum _{i=1}^{\infty }n_{i}\phi _{i}(t),\quad 0 |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4bfd8d3b1e436dbceb662b89f288903fa8306c93 height: height attribute not set width: width attribute not set description: {\displaystyle n_{i}=\int n(t)\phi _{i}(t)\,dt} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8669f2b4aacff8f167e1c3671175074a9c7e52bd height: height attribute not set width: width attribute not set description: {\displaystyle \{\phi _{i}{t}\}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6e99a3a6e64d39005be67e6cdc3622050b782bd5 height: height attribute not set width: width attribute not set description: {\displaystyle r_{n}(t,s)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5a3438932b073324caf252e7c0b3f9f19ac0791e height: height attribute not set width: width attribute not set description: {\displaystyle \int _{0}^{t}r_{n}(t,s)\phi _{i}(s)\,ds=\lambda _{i}\phi _{i}(t),\quad \operatorname {var} [n_{i}]=\lambda _{i}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1d81b202b95d7d95636909975570c6abfafbdd2b height: height attribute not set width: width attribute not set description: {\displaystyle s(t)=\sum _{i=1}^{\infty }s_{i}\phi _{i}(t),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cb30624a8829903bfd2f1856d95158f458d1a669 height: height attribute not set width: width attribute not set description: {\displaystyle s_{i}=\int _{0}^{t}s(t)\phi _{i}(t)\,dt} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bc9cf451529658ff57b32735650e16e7976e5d07 height: height attribute not set width: width attribute not set description: {\displaystyle x_{i}=\int _{0}^{t}x(t)\phi _{i}(t)\,dt=n_{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0f7dfa09505baf75100c13dcdea4a82562989b38 height: height attribute not set width: width attribute not set description: {\displaystyle n_{i}+s_{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/2370c3a347a97209764daee028ea3d60c8f0158d height: height attribute not set width: width attribute not set description: {\displaystyle {\overline {x}}=\{x_{1},x_{2},\dots \}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fef58cebf23adff9199f17325aefb5515fdca99d height: height attribute not set width: width attribute not set description: {\displaystyle n_{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/72fde940918edf84caf3d406cc7d31949166820f height: height attribute not set width: width attribute not set description: {\displaystyle \lambda _{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba8a662382c1f121e79e3f1d518c575cb6c42c6 height: height attribute not set width: width attribute not set description: {\displaystyle \{x_{i}\}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5106d08c991f9d03bb7bb076c831a0f33200d4bc height: height attribute not set width: width attribute not set description: {\displaystyle f_{h}[x(t)|0 |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fe92010d1d4ffef65d2cd419bc10ce6d6f6cfffc height: height attribute not set width: width attribute not set description: {\displaystyle \{x_{i}-s_{i}\}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/14a5bf34e1d35f9432e0a659d53f733ce651a83b height: height attribute not set width: width attribute not set description: {\displaystyle f_{k}[x(t)\mid 0 |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fa8f333d96981a0aa6998102d9e61edda873115e height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {l}}({\underline {x}})=\sum _{i=1}^{\infty }{\frac {2s_{i}x_{i}-s_{i}^{2}}{2\lambda _{i}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/23335ab5c27be1fc89ba54c166dc3fa429db66d3 height: height attribute not set width: width attribute not set description: {\displaystyle g=\sum _{i=1}^{\infty }s_{i}x_{i}\lambda _{i}>g_{0}\rightarrow k, |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1dcca9a86661d2323cbac8bcc097e1e46050498f height: height attribute not set width: width attribute not set description: {\displaystyle k(t)=\sum _{i=1}^{\infty }\lambda _{i}s_{i}\phi _{i}(t),0 |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5ed011f5adf8eb84cece33cceb4cbd9fa9e19342 height: height attribute not set width: width attribute not set description: {\displaystyle g=\int _{0}^{t}k(t)x(t)\,dt.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/42eae063dbc96346aef46c02bfdd25dc14b13b7a height: height attribute not set width: width attribute not set description: {\displaystyle \int _{0}^{t}r_{n}(t,s)k(s)\,ds=\sum _{i=1}^{\infty }\lambda _{i}s_{i}\int _{0}^{t}r_{n}(t,s)\phi _{i}(s)\,ds=\sum _{i=1}^{\infty }s_{i}\phi _{i}(t)=s(t),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/582b7e37766612546255a8f9cfeabc97a66653e3 height: height attribute not set width: width attribute not set description: {\displaystyle \int _{0}^{t}r_{n}(t,s)k(s)\,ds=s(t).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e63dcd07c4512f0bd19f0ce904aab5aed0fce7a7 height: height attribute not set width: width attribute not set description: {\displaystyle \int _{0}^{t}r_{n}(t-s)k(s)\,ds=s(t),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8251371e7fca6fdaefa66a12971399e5abbf9c72 height: height attribute not set width: width attribute not set description: {\displaystyle \int _{0}^{t}{\frac {n_{0}}{2}}\delta (t-s)k(s)\,ds=s(t)\rightarrow k(t)=cs(t),\quad 0 |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7b0ad9b5db9969a09214483f638b8e27c75c9bf0 height: height attribute not set width: width attribute not set description: {\displaystyle g=\int _{0}^{t}k(t)x(t)\,dt,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d28a73179b98c25cf752e7f06937466182b316c4 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\mathbf {e} [g\mid h]&=\int _{0}^{t}k(t)\mathbf {e} [x(t)\mid h]\,dt=0\\\mathbf {e} [g\mid k]&=\int _{0}^{t}k(t)\mathbf {e} [x(t)\mid k]\,dt=\int _{0}^{t}k(t)s(t)\,dt\equiv \rho \\\mathbf {e} [g^{2}\mid h]&=\int _{0}^{t}\int _{0}^{t}k(t)k(s)r_{n}(t,s)\,dt\,ds=\int _{0}^{t}k(t)\left(\int _{0}^{t}k(s)r_{n}(t,s)\,ds\right)=\int _{0}^{t}k(t)s(t)\,dt=\rho \\\operatorname {var} [g\mid h]&=\mathbf {e} [g^{2}\mid h]-(\mathbf {e} [g\mid h])^{2}=\rho \\\mathbf {e} [g^{2}\mid k]&=\int _{0}^{t}\int _{0}^{t}k(t)k(s)\mathbf {e} [x(t)x(s)]\,dt\,ds=\int _{0}^{t}\int _{0}^{t}k(t)k(s)(r_{n}(t,s)+s(t)s(s))\,dt\,ds=\rho +\rho ^{2}\\\operatorname {var} [g\mid k]&=\mathbf {e} [g^{2}|k]-(\mathbf {e} [g|k])^{2}=\rho +\rho ^{2}-\rho ^{2}=\rho \end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/83f269f59cba721507b641497ce938df10592caf height: height attribute not set width: width attribute not set description: {\displaystyle h:g\sim n(0,\rho )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4fb502352b7adeb845c12edbe94efa0a52205c26 height: height attribute not set width: width attribute not set description: {\displaystyle k:g\sim n(\rho ,\rho )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d15e9c2d2f4cfc0b041cce83a1efdba5e57210 height: height attribute not set width: width attribute not set description: {\displaystyle \alpha =\int _{g_{0}}^{\infty }n(0,\rho )\,dg=1-\phi \left({\frac {g_{0}}{\sqrt {\rho }}}\right).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/55fe11e3d9f34e6b40d694a64216ea6361484d71 height: height attribute not set width: width attribute not set description: {\displaystyle g_{0}={\sqrt {\rho }}\phi ^{-1}(1-\alpha ).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/faec25200333b5cc2165afa719418287a8aecf46 height: height attribute not set width: width attribute not set description: {\displaystyle \beta =\int _{g_{0}}^{\infty }n(\rho ,\rho )\,dg=\phi \left({\sqrt {\rho }}-\phi ^{-1}(1-\alpha )\right)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fe5863880fedccac365f79bf60a3346ea79ef87a height: height attribute not set width: width attribute not set description: {\displaystyle \rho =\int _{0}^{t}k(t)s(t)\,dt=\int _{0}^{t}s(t)^{2}\,dt=e.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/76cfbecc7fd63227b96a4f089c77436f6b62c7f5 height: height attribute not set width: width attribute not set description: {\displaystyle r_{n}(\tau )={\frac {bn_{0}}{4}}e^{-b|\tau |}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f3e9aa5c68e57d597d818eebb4bf4d601b1dfe32 height: height attribute not set width: width attribute not set description: {\displaystyle s_{n}(f)={\frac {n_{0}}{2(1+({\frac {w}{b}})^{2})}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/28edba5b514e18975735bd13aea883355796339b height: height attribute not set width: width attribute not set description: {\displaystyle h(f)=1+j{\frac {w}{b}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4cd801c7a19617d4da70d2fb228e2dac23cde1ad height: height attribute not set width: width attribute not set description: {\displaystyle h_{0}:y(t)=n(t)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fc8e575573deaaf333d98ecbe7393375a35622db height: height attribute not set width: width attribute not set description: {\displaystyle h_{1}:y(t)=n(t)+x(t),\quad 0 |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/629038a04725c88a606ce7b99c6580f5f4709186 height: height attribute not set width: width attribute not set description: {\displaystyle r_{x}(t,s)=e\{x(t)x(s)\}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b2091487683764070c1a1d8fe4a55e46259c5ef7 height: height attribute not set width: width attribute not set description: {\displaystyle x(t)=\sum _{i=1}^{\infty }x_{i}\phi _{i}(t),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d4d7bc54afdaeff2745be8330715d8a4d96a1d94 height: height attribute not set width: width attribute not set description: {\displaystyle x_{i}=\int _{0}^{t}x(t)\phi _{i}(t)\,dt} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3453efaaeb4bf73cb4ef295008d1c17efcff2121 height: height attribute not set width: width attribute not set description: {\displaystyle \phi _{i}(t)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5f7dbf325ff2b4c13cb1f6411708a6b249a7b44e height: height attribute not set width: width attribute not set description: {\displaystyle \int _{0}^{t}r_{x}(t,s)\phi _{i}(s)ds=\lambda _{i}\phi _{i}(t).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/af4a0955af42beb5f85aa05fb8c07abedc13990d height: height attribute not set width: width attribute not set description: {\displaystyle x_{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3f98059b8f7146860740552c3f232a4387c8abe1 height: height attribute not set width: width attribute not set description: {\displaystyle y_{i}=\int _{0}^{t}y(t)\phi _{i}(t)\,dt=\int _{0}^{t}[n(t)+x(t)]\phi _{i}(t)=n_{i}+x_{i},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8e93ce7e07f13392e95648cc76b09d83de258a51 height: height attribute not set width: width attribute not set description: {\displaystyle n_{i}=\int _{0}^{t}n(t)\phi _{i}(t)\,dt.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/04daf1cddcdc9dc8910258c1de8e201eb5db6ec6 height: height attribute not set width: width attribute not set description: {\displaystyle {\tfrac {1}{2}}n_{0}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/fe46c66c6eeaf0d15c449f9d72f15f1674e55131 height: height attribute not set width: width attribute not set description: {\displaystyle h_{0}:y_{i}=n_{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1f3714a0cee3697564ec0c2ddb1499b8c35247dc height: height attribute not set width: width attribute not set description: {\displaystyle h_{1}:y_{i}=n_{i}+x_{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a063d5e574c467dd8246f5f2278460f862a2086e height: height attribute not set width: width attribute not set description: {\displaystyle \lambda ={\frac {f_{y}\mid h_{1}}{f_{y}\mid h_{0}}}=ce^{-\sum _{i=1}^{\infty }{\frac {y_{i}^{2}}{2}}{\frac {\lambda _{i}}{{\tfrac {1}{2}}n_{0}({\tfrac {1}{2}}n_{0}+\lambda _{i})}}},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4d95aa8c7212baa6870a30a9306402a8db61d78f height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {l}}=\ln(\lambda )=k-\sum _{i=1}^{\infty }{\tfrac {1}{2}}y_{i}^{2}{\frac {\lambda _{i}}{{\frac {n_{0}}{2}}\left({\frac {n_{0}}{2}}+\lambda _{i}\right)}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e64a3a3890efea38a15655e476ca26acc1ac57a5 height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {x}}_{i}={\frac {\lambda _{i}}{{\frac {n_{0}}{2}}\left({\frac {n_{0}}{2}}+\lambda _{i}\right)}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d57be496fff95ee2a97ee43c7f7fe244b4dbf8ae height: height attribute not set width: width attribute not set description: {\displaystyle y_{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/08f3f155772491261806196461bd4de516bfac3b height: height attribute not set width: width attribute not set description: {\displaystyle {\mathcal {l}}=k+{\frac {1}{n_{0}}}\sum _{i=1}^{\infty }y_{i}{\widehat {x}}_{i}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/be3e2e735b7e3f5ad27257aeb1a76a3fd627e28b height: height attribute not set width: width attribute not set description: {\displaystyle f(t)=\sum f_{i}\phi _{i}(t),g(t)=\sum g_{i}\phi _{i}(t),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5e63450fd5a0ee92518a234880a8ecbde5b5cd03 height: height attribute not set width: width attribute not set description: {\displaystyle f_{i}=\int _{0}^{t}f(t)\phi _{i}(t)\,dt,\quad g_{i}=\int _{0}^{t}g(t)\phi _{i}(t)\,dt.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/0321f52240cc75eb52dc4dbc081428a99956ab77 height: height attribute not set width: width attribute not set description: {\displaystyle \sum _{i=1}^{\infty }f_{i}g_{i}=\int _{0}^{t}g(t)f(t)\,dt.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/5533ffca0956eb649b648cdc555bd55e267b731b height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {x}}(t\mid t)=\sum _{i=1}^{\infty }{\widehat {x}}_{i}\phi _{i}(t),\quad {\mathcal {l}}=k+{\frac {1}{n_{0}}}\int _{0}^{t}y(t){\widehat {x}}(t\mid t)\,dt.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/efc1a9380c5ae2d924984c4fc2ca7308fd17069c height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {x}}(t\mid t)=\int _{0}^{t}q(t,s)y(s)\,ds.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d05d7baacd5d8ea6a3d3aff4aed1163675348fef height: height attribute not set width: width attribute not set description: {\displaystyle \int _{0}^{t}q(t,s)r_{x}(s,t)\,ds+{\tfrac {n_{0}}{2}}q(t,\lambda )=r_{x}(t,\lambda ),0<\lambda |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/160b7c3266686f753dd12a8610adf531e9f7acfe height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {x}}(t\mid t)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/b6f49361bc240390a86ebb9f078e2478779ff140 height: height attribute not set width: width attribute not set description: {\displaystyle q(t,s)=h(t,s)+h(s,t)-\int _{0}^{t}h(\lambda ,t)h(s,\lambda )\,d\lambda } |
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https://login.wikimedia.org/wiki/special:centralautologin/start?type=1x1 height: 1 width: 1 description: no alt description found |
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http://en.wikipedia.org/static/images/footer/wikimedia-button.svg height: 29 width: 84 description: wikimedia foundation |
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http://en.wikipedia.org/w/resources/assets/poweredby_mediawiki.svg height: 31 width: 88 description: powered by mediawiki |
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