Basis function








In mathematics, a basis function is an element of a particular basis for a function space. Every continuous function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors.


In numerical analysis and approximation theory, basis functions are also called blending functions, because of their use in interpolation: In this application, a mixture of the basis functions provides an interpolating function (with the "blend" depending on the evaluation of the basis functions at the data points).




Contents






  • 1 Examples


    • 1.1 Polynomial bases


    • 1.2 Fourier basis




  • 2 References


  • 3 See also


  • 4 References





Examples



Polynomial bases


The base of a polynomial is the factored polynomial equation into a linear function.[1]



Fourier basis


Sines and cosines form an (orthonormal) Schauder basis for square-integrable functions. As a particular example, the collection:


{2sin⁡(2πnx)|n∈N}∪{2cos⁡(2πnx)|n∈N}∪{1}{displaystyle {{sqrt {2}}sin(2pi nx);|;nin mathbb {N} }cup {{sqrt {2}}cos(2pi nx);|;nin mathbb {N} }cup {1}}{{sqrt {2}}sin(2pi nx);|;nin mathbb {N} }cup {{sqrt {2}}cos(2pi nx);|;nin mathbb {N} }cup {1}

forms a basis for L2(0,1).



References



  • Ito, Kiyoshi (1993). Encyclopedic Dictionary of Mathematics (2nd ed.). MIT Press. p. 1141. ISBN 0-262-59020-4..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


See also












References




  1. ^ "Solutions of differential equations in a Bernstein polynomial basis". Journal of Computational and Applied Mathematics. 205 (1): 272–280. 2007-08-01. doi:10.1016/j.cam.2006.05.002. ISSN 0377-0427.








Comments

Post a Comment

Popular posts from this blog

Information security

Image
This article is part of a series on Information security Related security categories Internet security Cyberwarfare Computer security Mobile security Network security Threats Computer crime Vulnerability Eavesdropping Malware Spyware Ransomware Trojans Viruses Worms Rootkits Bootkits Keyloggers Screen scrapers Exploits Backdoors Logic bombs Payloads Denial of service Defenses Computer access control Application security Antivirus software Secure coding Secure by default Secure by design Secure operating systems Authentication Multi-factor authentication Authorization Data-centric security Encryption Firewall Intrusion detection system Mobile secure gateway Runtime application self-protection (RASP) v t e Information security , sometimes shortened to InfoSec , is the practice of preventing unauthorized access, use, disclosure, disruption, modification, inspection, recording or destruction of information. Th...
Read more

Volkswagen Group MQB platform

Image
The Volkswagen Group MQB platform is the company's strategy for shared modular design construction of its transverse, front-engine, front-wheel-drive layout (optional front-engine, four-wheel-drive layout) automobiles. Volkswagen spent roughly $60bn [1] developing this new platform and the cars employing it. The platform underpins a wide range of cars from the supermini class to the mid size SUV class. MQB allows Volkswagen to assemble any of its cars based on this platform across all of its MQB ready factories. This allows the Volkswagen group flexibility to shift production as needed between its different factories. Beginning in 2012, Volkswagen Group marketed the strategy under the code name MQB , which stands for Modularer Querbaukasten , translating from German to "Modular Transversal Toolkit" or "Modular Transverse Matrix". [2] [3] MQB is one strategy within VW's overall MB (Modularer Baukasten or modular matrix) program which also includes th...
Read more

刘萌萌

Image
body.skin-minerva .mw-parser-output table.infobox caption{text-align:center} 刘萌萌 女演员 罗马拼音 Liu Meng Meng 英文名 Cya 国籍   中国 民族 汉族 出生 ( 1989-10-16 ) 1989年10月16日 （ 29歲）   中国辽宁省大连市 职业 演员、艺人 语言 汉语 教育程度 沈阳音乐学院 北京电影学院表演系进修班 活跃年代 2011年至今 唱片公司 浙江永乐影视制作有限公司 经纪公司 浙江永乐影视制作有限公司 网站 刘萌萌 官方微博 刘萌萌 （1989年10月16日 － ），汉族，中国演员，2010年参加湖南卫视《芒果训练营》而出道至今，凭借电视剧《爱情公寓》“诺澜”一角走红。北京电影学院10级表演系（进修班）毕业。 目录 1 主要经历 2 影视作品 2.1 电视剧 2.2 电影 2.3 综艺节目 3 参考文献 4 外部链接 主要经历 1989年，出生于辽宁省大连市一个工人家庭； 2006年，考入沈阳音乐学院舞蹈系本科班学习； 2010年，参加湖南卫视《芒果训练营》演艺新秀比赛而出道 [1] ；進入北京电影学院表演系进修班 2011年，出演电视剧《隋唐演义》“琼花公主”一角色； 2012年，担任电视剧《爱情公寓》“诺澜”一角并开始走红 [2] ； 2013年，担任电视剧《利箭纵横》《利剑行动》 [3] 女主角“林筱雨” 2014年，参加江苏卫视真人秀栏目《花样年华》 [4] 备受瞩目 [5] 。 影视作品 电视剧 首播年份 剧名 角色 备注 2012 《爱情公寓3》 诺澜 2012 《利箭行动》 林筱雨 2013 《隋唐演义》 琼花公主 2013 《决战燕子门》 马晓婉 2014 《利箭纵横》 林筱雨 2014 《屌丝日记》 林洛霏 網絡劇 2014 《爱情公寓4》 诺澜 ...
Read more