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Orthogonal polynomials

∫ a b p m ⁢ ( x ) ⁢ p n ⁢ ( x ) ⁢ w ⁢ ( x ) ⁢ 𝑑 x = 0 , m ≠ n formulae-sequence superscript subscript 𝑎 𝑏 subscript 𝑝 𝑚 𝑥 subscript 𝑝 𝑛 𝑥 𝑤 𝑥 differential-d 𝑥 0 𝑚 𝑛 {\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 }

Constraint(s): m , n ∈ { 0 , 1 , 2 , … } 𝑚 𝑛 0 1 2 … {\displaystyle{\displaystyle{\displaystyle m,n\in\{0,1,2,\ldots\}}}}


0 < ∫ a b x 2 ⁢ n ⁢ w ⁢ ( x ) ⁢ 𝑑 x < ∞   for all formulae-sequence 0 superscript subscript 𝑎 𝑏 superscript 𝑥 2 𝑛 𝑤 𝑥 differential-d 𝑥 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} }

Constraint(s): n ∈ { 0 , 1 , 2 , … } 𝑛 0 1 2 … {\displaystyle{\displaystyle{\displaystyle n\in\{0,1,2,\ldots\}}}}


0 < ∫ a b x 2 ⁢ n ⁢ 𝑑 α ⁢ ( x ) < ∞   for all formulae-sequence 0 superscript subscript 𝑎 𝑏 superscript 𝑥 2 𝑛 differential-d 𝛼 𝑥 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} }

Constraint(s): n ∈ { 0 , 1 , 2 , … } 𝑛 0 1 2 … {\displaystyle{\displaystyle{\displaystyle n\in\{0,1,2,\ldots\}}}}


∑ x ∈ X p m ⁢ ( x ) ⁢ p n ⁢ ( x ) ⁢ w x = 0 , m ≠ n formulae-sequence subscript 𝑥 𝑋 subscript 𝑝 𝑚 𝑥 subscript 𝑝 𝑛 𝑥 subscript 𝑤 𝑥 0 𝑚 𝑛 {\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 }

Constraint(s): m , n ∈ { 0 , 1 , 2 , … } 𝑚 𝑛 0 1 2 … {\displaystyle{\displaystyle{\displaystyle m,n\in\{0,1,2,\ldots\}}}}


∑ x ∈ X p m ⁢ ( x ) ⁢ p n ⁢ ( x ) ⁢ w x = 0 , m ≠ n formulae-sequence subscript 𝑥 𝑋 subscript 𝑝 𝑚 𝑥 subscript 𝑝 𝑛 𝑥 subscript 𝑤 𝑥 0 𝑚 𝑛 {\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 }

Constraint(s): m , n ∈ { 0 , 1 , 2 , … , N } 𝑚 𝑛 0 1 2 … 𝑁 {\displaystyle{\displaystyle{\displaystyle m,n\in\{0,1,2,\ldots,N\}}}}


σ n = ∫ a b { p n ⁢ ( x ) } 2 ⁢ w ⁢ ( x ) ⁢ 𝑑 x subscript 𝜎 𝑛 superscript subscript 𝑎 𝑏 superscript subscript 𝑝 𝑛 𝑥 2 𝑤 𝑥 differential-d 𝑥 {\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 }

Constraint(s): n = 0 , 1 , 2 , … 𝑛 0 1 2 … {\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots}}}


σ n = ∑ x ∈ X { p n ⁢ ( x ) } 2 ⁢ w x subscript 𝜎 𝑛 subscript 𝑥 𝑋 superscript subscript 𝑝 𝑛 𝑥 2 subscript 𝑤 𝑥 {\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 }

Constraint(s): n = 0 , 1 , 2 , … 𝑛 0 1 2 … {\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots}}}


σ n = ∑ x ∈ X { p n ⁢ ( x ) } 2 ⁢ w x subscript 𝜎 𝑛 subscript 𝑥 𝑋 superscript subscript 𝑝 𝑛 𝑥 2 subscript 𝑤 𝑥 {\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 }

Constraint(s): n = 0 , 1 , 2 , … , N 𝑛 0 1 2 … 𝑁 {\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots,N}}}


p n ⁢ ( x ) = k n ⁢ x n + lower order terms subscript 𝑝 𝑛 𝑥 subscript 𝑘 𝑛 superscript 𝑥 𝑛 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} }

Constraint(s): n = 0 , 1 , 2 , … 𝑛 0 1 2 … {\displaystyle{\displaystyle{\displaystyle n=0,1,2,\ldots}}}


δ m , n := { 0 , m ≠ n , 1 , m = n , assign Kronecker-delta 𝑚 𝑛 cases 0 𝑚 𝑛 1 𝑚 𝑛 {\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. }

Constraint(s): m , n ∈ { 0 , 1 , 2 , … } 𝑚 𝑛 0 1 2 … {\displaystyle{\displaystyle{\displaystyle m,n\in\{0,1,2,\ldots\}}}}


∫ a b p m ⁢ ( x ) ⁢ p n ⁢ ( x ) ⁢ w ⁢ ( x ) ⁢ 𝑑 x = σ n ⁢ δ m , n superscript subscript 𝑎 𝑏 subscript 𝑝 𝑚 𝑥 subscript 𝑝 𝑛 𝑥 𝑤 𝑥 differential-d 𝑥 subscript 𝜎 𝑛 Kronecker-delta 𝑚 𝑛 {\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} }

Constraint(s): m , n ∈ { 0 , 1 , 2 , … } 𝑚 𝑛 0 1 2 … {\displaystyle{\displaystyle{\displaystyle m,n\in\{0,1,2,\ldots\}}}}


∑ x ∈ X p m ⁢ ( x ) ⁢ p n ⁢ ( x ) ⁢ w x = σ n ⁢ δ m , n subscript 𝑥 𝑋 subscript 𝑝 𝑚 𝑥 subscript 𝑝 𝑛 𝑥 subscript 𝑤 𝑥 subscript 𝜎 𝑛 Kronecker-delta 𝑚 𝑛 {\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} }

Constraint(s): m , n ∈ { 0 , 1 , 2 , … , N } 𝑚 𝑛 0 1 2 … 𝑁 {\displaystyle{\displaystyle{\displaystyle m,n\in\{0,1,2,\ldots,N\}}}}


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