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Orthogonal polynomials - Revision history
2024-03-19T11:43:13Z
Revision history for this page on the wiki
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2017-03-05T22:34:28Z
<p>DRMF</p>
<p><b>New page</b></p><div>{{DISPLAYTITLE:Orthogonal polynomials}}<br />
<div id="drmf_head"><br />
<div id="alignleft"> << [[Orthogonal Polynomials|Orthogonal Polynomials]] </div><br />
<div id="aligncenter"> [[Orthogonal_Polynomials#Orthogonal polynomials|Orthogonal polynomials]] </div><br />
<div id="alignright"> [[The gamma and beta function|The gamma and beta function]] >> </div><br />
</div><br />
<br />
== Orthogonal polynomials ==<br />
<br />
<math id="KLS:01.01:01">{\displaystyle <br />
\int_a^bp_m(x)p_n(x)w(x)\,dx=0,\quad m\neq n<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle m,n\in\{0,1,2,\ldots\}}</math></div><br /><br />
<math id="KLS:01.01:02">{\displaystyle <br />
0<\int_a^bx^{2n}w(x)\,dx<\infty\quad\textrm{for all}<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle n\in\{0,1,2,\ldots\}}</math></div><br /><br />
<math id="KLS:01.01:03">{\displaystyle <br />
0<\int_a^bx^{2n}\,d\alpha(x)<\infty\quad\textrm{for all}<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle n\in\{0,1,2,\ldots\}}</math></div><br /><br />
<math id="KLS:01.01:04">{\displaystyle <br />
\sum_{x\in X}p_m(x)p_n(x)w_x=0,\quad m\neq n<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle m,n\in\{0,1,2,\ldots\}}</math></div><br /><br />
<math id="KLS:01.01:05">{\displaystyle <br />
\sum_{x\in X}p_m(x)p_n(x)w_x=0,\quad m\neq n<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle m,n\in\{0,1,2,\ldots,N\}}</math></div><br /><br />
<math id="KLS:01.01:06">{\displaystyle <br />
\sigma_n=\int_a^b\left\{p_n(x)\right\}^2w(x)\,dx<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br /><br />
<math id="KLS:01.01:07">{\displaystyle <br />
\sigma_n=\sum_{x\in X}\left\{p_n(x)\right\}^2w_x<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br /><br />
<math id="KLS:01.01:08">{\displaystyle <br />
\sigma_n=\sum_{x\in X}\left\{p_n(x)\right\}^2w_x<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots,N}</math></div><br /><br />
<math id="KLS:01.01:09">{\displaystyle <br />
p_n(x)=k_nx^n+\,\textrm{lower order terms}<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br /><br />
<math id="KLS:01.01:10">{\displaystyle <br />
\,\Kronecker{m}{n}:=\left\{\begin{array}{ll}0, &m\neq n,\\[5mm]<br />
1, & m=n,\end{array}\right.<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle m,n\in\{0,1,2,\ldots\}}</math></div><br /><br />
<math id="KLS:01.01:11">{\displaystyle <br />
\int_a^bp_m(x)p_n(x)w(x)\,dx=\sigma_n\,\Kronecker{m}{n}<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle m,n\in\{0,1,2,\ldots\}}</math></div><br /><br />
<math id="KLS:01.01:12">{\displaystyle <br />
\sum_{x\in X}p_m(x)p_n(x)w_x=\sigma_n\,\Kronecker{m}{n}<br />
}</math><br />
<div align="right">Constraint(s): <math>{\displaystyle m,n\in\{0,1,2,\ldots,N\}}</math></div><br /><br />
<div id="drmf_foot"><br />
<div id="alignleft"> << [[Orthogonal Polynomials|Orthogonal Polynomials]] </div><br />
<div id="aligncenter"> [[Orthogonal_Polynomials#Orthogonal polynomials|Orthogonal polynomials]] </div><br />
<div id="alignright"> [[The gamma and beta function|The gamma and beta function]] >> </div><br />
</div></div>
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