â« a b p m ⢠( x ) ⢠p n ⢠( x ) ⢠w ⢠( x ) ⢠ð x = 0 , m â n formulae-sequence superscript subscript ð ð subscript ð ð ð¥ subscript ð ð ð¥ ð¤ ð¥ differential-d ð¥ 0 ð ð {\displaystyle{\displaystyle{\displaystyle\int_{a}^{b}p_{m}(x)p_{n}(x)w(x)\,dx% =0,\quad m\neq n}}} {\displaystyle \int_a^bp_m(x)p_n(x)w(x)\,dx=0,\quad m\neq n }
0 < â« a b x 2 ⢠n ⢠w ⢠( x ) ⢠ð x < â â for all formulae-sequence 0 superscript subscript ð ð superscript ð¥ 2 ð ð¤ ð¥ differential-d ð¥ for all {\displaystyle{\displaystyle{\displaystyle 0<\int_{a}^{b}x^{2n}w(x)\,dx<\infty% \quad\textrm{for all}}}} {\displaystyle 0<\int_a^bx^{2n}w(x)\,dx<\infty\quad\textrm{for all} }
0 < â« a b x 2 ⢠n ⢠ð α ⢠( x ) < â â for all formulae-sequence 0 superscript subscript ð ð superscript ð¥ 2 ð differential-d ð¼ ð¥ for all {\displaystyle{\displaystyle{\displaystyle 0<\int_{a}^{b}x^{2n}\,d\alpha(x)<% \infty\quad\textrm{for all}}}} {\displaystyle 0<\int_a^bx^{2n}\,d\alpha(x)<\infty\quad\textrm{for all} }
â x â X p m ⢠( x ) ⢠p n ⢠( x ) ⢠w x = 0 , m â n formulae-sequence subscript ð¥ ð subscript ð ð ð¥ subscript ð ð ð¥ subscript ð¤ ð¥ 0 ð ð {\displaystyle{\displaystyle{\displaystyle\sum_{x\in X}p_{m}(x)p_{n}(x)w_{x}=0% ,\quad m\neq n}}} {\displaystyle \sum_{x\in X}p_m(x)p_n(x)w_x=0,\quad m\neq n }
Ï n = â« a b { p n ⢠( x ) } 2 ⢠w ⢠( x ) ⢠ð x subscript ð ð superscript subscript ð ð superscript subscript ð ð ð¥ 2 ð¤ ð¥ differential-d ð¥ {\displaystyle{\displaystyle{\displaystyle\sigma_{n}=\int_{a}^{b}\left\{p_{n}(% x)\right\}^{2}w(x)\,dx}}} {\displaystyle \sigma_n=\int_a^b\left\{p_n(x)\right\}^2w(x)\,dx }
Ï n = â x â X { p n ⢠( x ) } 2 ⢠w x subscript ð ð subscript ð¥ ð superscript subscript ð ð ð¥ 2 subscript ð¤ ð¥ {\displaystyle{\displaystyle{\displaystyle\sigma_{n}=\sum_{x\in X}\left\{p_{n}% (x)\right\}^{2}w_{x}}}} {\displaystyle \sigma_n=\sum_{x\in X}\left\{p_n(x)\right\}^2w_x }
p n ⢠( x ) = k n ⢠x n + lower order terms subscript ð ð ð¥ subscript ð ð superscript ð¥ ð lower order terms {\displaystyle{\displaystyle{\displaystyle p_{n}(x)=k_{n}x^{n}+\,\textrm{lower% order terms}}}} {\displaystyle p_n(x)=k_nx^n+\,\textrm{lower order terms} }
δ m , n := { 0 , m â n , 1 , m = n , assign Kronecker-delta ð ð cases 0 ð ð 1 ð ð {\displaystyle{\displaystyle{\displaystyle\,\delta_{m,n}:=\left\{\begin{array}% []{ll}0,&m\neq n,\\ 1,&m=n,\end{array}\right.}}} {\displaystyle \,\Kronecker{m}{n}:=\left\{\begin{array}{ll}0, &m\neq n,\[5mm] 1, & m=n,\end{array}\right. }
â« a b p m ⢠( x ) ⢠p n ⢠( x ) ⢠w ⢠( x ) ⢠ð x = Ï n ⢠δ m , n superscript subscript ð ð subscript ð ð ð¥ subscript ð ð ð¥ ð¤ ð¥ differential-d ð¥ subscript ð ð Kronecker-delta ð ð {\displaystyle{\displaystyle{\displaystyle\int_{a}^{b}p_{m}(x)p_{n}(x)w(x)\,dx% =\sigma_{n}\,\delta_{m,n}}}} {\displaystyle \int_a^bp_m(x)p_n(x)w(x)\,dx=\sigma_n\,\Kronecker{m}{n} }
â x â X p m ⢠( x ) ⢠p n ⢠( x ) ⢠w x = Ï n ⢠δ m , n subscript ð¥ ð subscript ð ð ð¥ subscript ð ð ð¥ subscript ð¤ ð¥ subscript ð ð Kronecker-delta ð ð {\displaystyle{\displaystyle{\displaystyle\sum_{x\in X}p_{m}(x)p_{n}(x)w_{x}=% \sigma_{n}\,\delta_{m,n}}}} {\displaystyle \sum_{x\in X}p_m(x)p_n(x)w_x=\sigma_n\,\Kronecker{m}{n} }