User:Egm6936.f10/Orthogonal polynomials

Xiu 2010, Chapter 3, p.25

Polynomials, especially orthogonal polynomials, are widely used in modern approximation theory. It's of great value to review some basis about the orthogonal polynomial. For the purpose of simplicity, we only focus on univariate case here.

=Orthogonal polynomial=

A general $$\displaystyle n$$ - degree polynomial of $$\displaystyle x$$ can be expressed in the form as
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$$\displaystyle P(x)=a_{n}x^{n} + a_{n-1}x^{n-1} + ... + a_{1}x^{1} + a_{0}x^{0} $$ (ab1) In above expression, the first coefficient $$a_{n}$$ is termed leading coefficient. When $$a_{n} = 1$$, we call the polynomial as monic polynomial.
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Weighted inner product of two real functions
The inner product of two real functions is defined as
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$$\displaystyle \left \langle f(x), g(x) \right \rangle = \int_{S}f(x)g(x)dx $$ (ab2) where $$\displaystyle S$$ is the supporting domain. And the weighted inner product of two real functions is defined as
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$$\displaystyle \left \langle f(x), g(x) \right \rangle_{\omega} = \int_{S}f(x)g(x)\omega(x)dx $$ (ab3) where $$\displaystyle \omega(x)$$ is the weight function.
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Orthogonal polynomial system
A orthogonal system of polynomials is a family of polynomials satisfy that
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$$\displaystyle \int_{S}P_m(x)P_n(x)d\alpha(x) = \gamma_n\delta_{mn}, \;\;\;\; m,n \in {0, 1, 2, ...} $$ (ab4) with respect to some real positive measure $$\displaystyle \alpha $$. Where $$\displaystyle \delta_{mn} $$ is the Kronecker delta function, $$\displaystyle S $$ is the support of $$\displaystyle \alpha $$, $$\displaystyle \gamma_{n} $$ is the positive normalization constants. Apparently,
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$$\displaystyle \gamma_n = \int_{S}P_n(x)^2d\alpha(x), \;\;\;\; n \in {0, 1, 2, ...} $$ (ab5) If $$\displaystyle \gamma_{n} = 1$$, the system is orthonormal.
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Any orthogonal system can be made a orthonormal system by dividing $$\displaystyle P_{n} $$ by $$\displaystyle \gamma_{n} $$, that's $$\displaystyle \widetilde{P_{n}} = P_{n}/\gamma_{n} $$. $$\displaystyle \widetilde{P_{n}}$$ is a orthonormal system.

Usually, the measure $$\displaystyle \alpha $$ can either have a density $$\displaystyle \omega(x) $$ for continuous case, or weight $$\displaystyle \omega_i $$ at points $$\displaystyle x_i $$ for discrete case. The orthogonal relation then becomes
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$$\displaystyle \int_{S}P_m(x)P_n(x)\omega(x)dx = \gamma_n\delta_{mn}, \;\;\;\; m,n \in {0, 1, 2, ...} $$ (ab6) for continuous case, and
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$$\displaystyle \sum_{i}P_m(x_i)P_n(x_i)\omega_i = \gamma_n\delta_{mn}, \;\;\;\; m,n \in {0, 1, 2, ...} $$ (ab7) for discrete case. Utilizing the definition of weighted inner product of two real functions, we can rewrite the orthogonal relation as
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$$\displaystyle \left \langle P_m(x), P_n(x) \right \rangle_{w} = \gamma_n\delta_{mn}, \;\;\;\; m,n \in {0, 1, 2, ...} $$ (ab8) where
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$$\displaystyle \gamma_n = \left \langle P_n(x), P_n(x) \right \rangle_{w} = \left \| P_n(x) \right \|_{\omega}^2, \;\;\;\; n \in {0, 1, 2, ...} $$ (ab9)
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Three-term recurrence relation
All orthogonal polynomials satisfy a three-term recurrence relation. And the general recurrence relation is
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$$\displaystyle -xP_{n}(x) = a_{n}P_{n+1}(x) + b_{n}P_{n}(x) + c_{n}P_{n-1}(x), n \geq 1 $$ (ab10)
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Certain restriction is applied to some terms in above equation. These restrictions are $$\displaystyle a_n \neq 0$$, $$\displaystyle c_n \neq 0$$, $$\displaystyle c_n / a_{n-1} > 0$$, $$\displaystyle P_{-1}(x) = 0$$ and $$\displaystyle P_0(x) = 1$$.

Some useful orthogonal polynomials
Before proceeding to orthogonal polynomials, we introduce the Hypergeometric series(Wiki) first. Hypergeometric series is a series that the ratio of two adjacent terms is a simple function which only depends on the index.

 you can look at Battin 1999 Astrodynamics for explanation on hypergeometric series. Egm6322.s12 16:12, 5 January 2012 (UTC)

The Pochhammer symbol(Wiki)$$\displaystyle (a)^n$$ is defined as
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$$\displaystyle (a)^n = \left\{\begin{matrix} a, & n = 0 \\ a(a+1)...(a+n-1), & n = 1,2,... \end{matrix}\right. $$ (ab11) If $$\displaystyle a \in \mathbb N$$ is a positive integer, then
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$$\displaystyle (a)^n = \frac{(a+n-1)!}{(a-1)!}, \;\;\; n > 0 $$ (ab12) For general $$\displaystyle a \in \mathbb R$$
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$$\displaystyle (a)^n = \frac{\Gamma(a+n)}{\Gamma(a)}, \;\;\; n > 0 $$ (ab13) where $$\displaystyle \Gamma (\;)$$ is Gamma function(Wiki), an extension of factorial function. The generalized Hypergeometric series $$\displaystyle {}_{s}F_{r}$$is defined in terms of Pochhammer symbol as
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$$\displaystyle {}_{s}F_{r}(a_1,a_2,...,a_s;b_1,b_2,...,b_r;z) = \sum_{n=0}^{+\infty}\frac{(a_1)^n...(a_s)^n}{(b_1)^n...(b_r)^n}\frac{z^n}{n!} $$ (ab14) where $$\displaystyle b_i \neq 0,-1,-2,...$$. When $$\displaystyle a_i $$ is a negative integer, the series terminates because $$\displaystyle (-k)^n = 0$$. For example,
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$$\displaystyle {}_{0}F_{0}(\;;\;;z) = \sum_{n=0}^{+\infty}\frac{z^n}{n!} $$ (ab15)
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Legendre polynomials
Historically, Legendre polynomial(Wiki) is derived by Adrien-Marie Legendre(Wiki) when solving Laplace's equation(Wiki) in spherical coordinate system. The domain of the Legendre polynomials presented here is $$\displaystyle I = [-1, 1] $$. Legendre polynomials can be presented either using Hypergeometric series or Rodrigues' formula(Wiki) as In Hypergeometric series:
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$$\displaystyle P_n(x) = {}_{2}F_{n}(-n, n+1; 1; \frac{1-x}{2}) $$ (ab16) In Rodrigues' formular:
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$$\displaystyle P_n(x) = \frac{1}{2^nn!}\frac{dx^n}{d^n}[(x^2 - 1)^n] $$ (ab17)
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And the three-term recurrence relation is
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$$\displaystyle P_{n+1}(x) = \frac{2n+1}{n+1}xP_{n}(x) - \frac{n}{n+1}P_{n-1}(x), \;\;\; n > 0 $$ (ab18)
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The weighted inner product with weight function $$\displaystyle \omega(x) = 1$$ is
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$$\displaystyle \int_{-1}^{+1}P_n(x)P_m(x)dx = \frac{2}{2n+1}\delta_{mn} $$ (ab19) Using above recurrence relation, the first four Legendre polynomials can be derived as $$\displaystyle P_{0}(x) = 1$$ $$\displaystyle P_{1}(x) = x$$ $$\displaystyle P_{2}(x) = \frac{3}{2}x^2 - \frac{1}{2}$$ $$\displaystyle P_{3}(x) = \frac{5}{2}x^3 - \frac{3}{2}x$$ The first five Legendre polynomials
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Hermite polynomials
The Hermite polynomials(Wiki), which arise in Probability(Wiki), is a family of orthogonal polynomials named after Charles Hermite(Wiki). There are two standard version of Hermite polynomials: the probabilists(Wiki)' Hermite polynomials and the physicists(Wiki)' Hermite polynomials. The Hermite polynomials presented here are of the former version and were defined on $$\displaystyle I = \mathbb R $$. The Hermite polynomials expressed in the Hypergeometric series as
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$$\displaystyle H_n(x) = {\sqrt{2}x}^n {}_2F_0 (-\frac{n}{n};-\frac{n-1}{2};-\frac{2}{x^2}) $$ (ab20) The three-term recurrence relation is
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$$\displaystyle H_{n+1}(x) = xH_{n}(x) - nH_{n-1}(x), \;\;\; n > 0 $$ (ab21) The weighted inner product with weight function $$\displaystyle \omega(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$$ is
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$$\displaystyle \int_{-\infty}^{+\infty}H_n(x)H_m(x)\omega(x)dx = n!\delta_{mn} $$ (ab22) Using above recurrence relation, we can have the first four Hermite polynomials as $$\displaystyle H_{0}(x) = 1$$ $$\displaystyle H_{1}(x) = x$$ $$\displaystyle H_{2}(x) = x^2 - 1$$ $$\displaystyle H_{3}(x) = x^3 - 3x$$
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The first six (probabilists') Hermite polynomials

Laguerre polynomials
The Laguerre polynomials(Wiki) is a family of orthogonal polynomials named after Edmond Laguerre(Wiki). It's the canonical solution of Laguerre's equation:
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$$\displaystyle x\,y'' + (1 - x)\,y' + n\,y = 0\, $$ (ab23) The domain of the Laguerre polynomial presented here is $$\displaystyle I = [0, \infty) $$. Generalizing above second-order linear differential equation as
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$$\displaystyle x\,y'' + (\alpha + 1 - x)\,y' + n\,y = 0\, $$ (ab24) and solving it, we can have the generalized Laguerre polynomial as In Hypergeometric series:
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$$\displaystyle L_n^{\alpha}(x) = \frac{(\alpha + 1)n}{n!} {}_{1}F_{1}(-n; \alpha+1; x ),\;\;\; \alpha > -1 $$ (ab25) In Rodrigues' formular:
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$$\displaystyle L_{n}^{\alpha}(x) = \frac{x^{-\alpha}e^x}{n!}\frac{dx^n}{d^n}(e^{-x}x^{n+\alpha}) $$ (ab26) The three-term recurrence relation is
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$$\displaystyle (n+1)L_{n+1}^{\alpha}(x) = (-x+2n+\alpha+1)L_{n}^{\alpha}(x) - (n+\alpha)L_{n-1}^{\alpha}(x), \;\;\; n > 0 $$ (ab27) The weighted inner product with weight function $$\displaystyle \omega(x) = e^{-x}x^{\alpha}$$ is
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$$\displaystyle \int_{0}^{+\infty}L_n^{\alpha}(x)H_m^{\alpha}(x)\omega(x)dx = \frac{\Gamma(n+\alpha+1)}{n!}\delta_{mn} $$ (ab28) Setting $$\displaystyle \alpha = 0 $$, we obtain the simple Laguerre polynomials.
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$$\displaystyle L_{n}(x) = L_{n}^{0}(x) $$ (ab29) The first four simple Laguerre polynomials are $$\displaystyle L_{0}(x) = 1$$ $$\displaystyle L_{1}(x) = -x + 1$$ $$\displaystyle L_{2}(x) = \frac{1}{2}x^2 - 2x + 1$$ $$\displaystyle L_{3}(x) = -\frac{1}{6}x^3 - \frac{3}{2}x^2 - 3x + 1$$
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The first six Laguerre polynomials

=Polynomial approximation=

We provide the classical approximation theory here. Before giving the theorem, we define $$\displaystyle \mathbb P_n$$ as the linear at most n-degree polynomials space.
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$$\displaystyle \mathbb P_n = span\{x^k, k=0,1,...,n\} $$ (ab30) 【Weierstrass theorem 】: Let $$\displaystyle I $$ be a closed interval and $$\displaystyle f \in C^{0}(I) $$, there will always exists $$\displaystyle n \in \mathbb N $$ and $$\displaystyle p \in \mathbb P_n $$, for any $$\displaystyle \epsilon > 0 $$, such that
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$$\displaystyle \left | f(x)- p(x) \right | < \epsilon, \;\;\; \forall x \in I $$ (ab31)
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=Polynomial projection=

We define the weighted $$\displaystyle L^2$$ space as
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$$\displaystyle L_{\omega}^2(x) := \left \{ v: I \rightarrow \mathbb P | \int_{I}v^2(x)\omega(x)dx < +\infty \right \} $$ (ab32) where $$\displaystyle \omega (x)$$ is the weight function, and $$\displaystyle x \in I$$, $$\displaystyle I$$ is a general interval. And the weighted inner product is
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$$\displaystyle \left \langle f(x),g(x) \right \rangle_{L_{\omega}^2(I)} = \int_{I}f(x)g(x)\omega(x)dx,\;\;\; f(x),g(x) \in L_{\omega}^2(I) $$ (ab33) The norm is
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$$\displaystyle \left \| f(x) \right \|_{L_{\omega}^2(I)} = \left ( \int_{I}f(x)^2\omega(x)dx \right )^{1/2} $$ (ab34)
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We define projection operator$$\displaystyle P_{N}:L_{\omega}^2(I) \rightarrow \mathbb P_{N}$$.

Orthogonal projection
Let $$\displaystyle N$$ is a fixed nonnegative integer and $$\displaystyle \phi_{k}(x), \; k = 0,1,...,N$$ is orthogonal polynomials of degree at most $$\displaystyle N$$ with the weight function $$\displaystyle \omega (x)$$. For any function $$\displaystyle f \in L_{\omega}^2(I)$$
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$$\displaystyle P_{N}f := \sum_{k = 0}^{N}\widetilde{f_k}\phi_k(x) $$ (ab35) where
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$$\displaystyle \widetilde{f_k} := \frac{1}{\left \| \phi_k \right \|_{L_{\omega}^2}^2}\left \langle f,\phi_k \right \rangle_{L_{\omega}^2},\;\;\; 0\leq k \leq N $$ (ab36)
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Convergence rate
The rate of convergence depends on the generality of the interpolated function $$\displaystyle f$$ and the type of interpolating orthogonal bases $$\displaystyle \{ \phi_k \}$$.

When the function $$\displaystyle f$$ is not analytic, the convergence rate is no longer faster than the algebraic rate. For discontinuous functions, the convergence rate deteriorates significantly.

Gibbs phenomenon
Gibbs phenomenon(Wiki) With the increase of the number of orthogonal bases employed, the oscillations will not disappear, which is named the Gibbs phenomenon.

<div style="width: 80%; margin-left: auto; margin-right: auto; padding: 4px; border: 2px solid #FF0000; text-align: left;"> How can we cure the Gibbs phenomenon? Hylon.Chen 13:37, 19 January 2012 (UTC).

Not easy to cure; just use more and more terms in the series. you may want to add some screenshots from Wolfram Demo Projects. Egm6322.s12 15:35, 26 January 2012 (UTC) Fewer terms used in piecewise approximating discontinuous function using Fourier series

More terms used in piecewise approximating discontinuous function using Fourier series

Courtesy: Above two figures are screenshotted from Wolfram Mathworld.

Discrete projection
For a given function $$\displaystyle f \in L_{\omega}^2 (I)$$, the discrete projection is defined as
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$$\displaystyle I_{N}f := \sum_{k = 0}^{N}\widehat{f_k}\phi_k(x) $$ (ab37) with
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$$\displaystyle \widehat{f_k} = \frac{1}{\left \| \phi_k \right \|_{L_{\omega}^2}^2}\sum_{k=i}^{q}f_k(x^{(i)})\phi_k(x^{(i)})\omega_i,\;\;\; 0\leq k \leq N $$ (ab38) where $$\displaystyle x^{(i)}$$ are integration points and $$\displaystyle w_{i}$$ are the corresponding weights. With certain integration technique(for instance, Gauss quadrature) used, the coefficient $$\displaystyle \widehat f_k$$ approximates the coefficient $$\displaystyle \widetilde{f_k}$$ in the continuous orthogonal projection. Note: The only difference between continuous orthogonal projection and discrete projection is the integration technique approximates the exact continuous integration of $$\displaystyle \widetilde{f_k}$$.
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=Polynomial interpolation=

Polynomial Interpolation(Wiki): Given $$\displaystyle n + 1$$ pairs of $$\displaystyle (x_i,y_i)$$, find a function $$\displaystyle G = G(x) $$ such that $$\displaystyle G(x_i) = y_i$$ for all given $$\displaystyle (x_i,y_i)$$. $$\displaystyle \{x_i\} $$ is the interpolation nodes, $$\displaystyle G $$ is the interpolating polynomial. $$\displaystyle G $$ can be algebraic polynomial, trigonometrical polynomial, piecewise polynomial (local polynomial) and rational polynomial.

Existence
Given a set of points $$\displaystyle (x_i,y_i)$$, we can always find a function $$\displaystyle G(x) $$, such that $$\displaystyle G(x_i) = y_i$$ for all given $$\displaystyle (x_i,y_i)$$.

Uniqueness
For any given set of points $$\displaystyle (x_i,y_i)$$, we can find a unique function $$\displaystyle G(x) $$, such that $$\displaystyle G(x_i) = y_i$$ for all given $$\displaystyle (x_i,y_i)$$.

Interpolation error
Assuming $$\displaystyle f$$ is the given function to be interpolated with the range of $$\displaystyle I$$, where $$\displaystyle x_0,x_1,...,x_n$$ are the distinct nodes within the range, and $$\displaystyle \Pi_nf$$ be the interpolating polynomial of $$\displaystyle nth$$ degree.

【Theorem】: The interpolation error at point $$\displaystyle x$$ with the interval $$\displaystyle I$$ is
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$$\displaystyle E_{N}(x) = f(x)-\Pi_nf(x) = \frac{f^{n+1}(\xi)}{(n+1)!}q_{n+1}(x) $$ (ab39)
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where $$\displaystyle I_x $$ is the smallest interval including $$\displaystyle x,x_0,x_1,...,x_n$$ and $$\displaystyle q_{n+1}(x) = \Pi^{n+1}_{i=0}(x-x_i)$$ is the nodal Lagrange polynomial of degree $$\displaystyle n+1 $$.

For detailed proof of this theorem, one can refer to [[media:nm1.s11.mtg11.djvu|NM1S11 - Mtg 11.2]].

Runge phenomenon
Runge phenomenon (Wiki): When using high-degree polynomials in a interpolation with evenly distributed nodes, $$\displaystyle q_{n+1}(x) = \Pi^{n+1}_{i=0}(x-x_i)$$ will performs rather wildly near the endpoint nodes. This leads to bad convergence at the endpoint nodes. This is called the Runge phenomenon. To cure this phenomenon, one can either use piecewise low-degree polynomials or non-uniformly distributed nodes for high-degree polynomials. Also refer to [[media:nm1.s11.mtg21.djvu|NM1S11 - Mtg 21]].

=Zeros of orthogonal polynomials and quadrature=

【Theorem】: For $$\displaystyle n \geq 1$$, orthogonal polynomials $$\displaystyle P_n, \;\;n \in \mathbb N$$ has exactly $$\displaystyle n $$ real different zeros in $$\displaystyle I $$.

<div style="width: 80%; margin-left: auto; margin-right: auto; padding: 4px; border: 2px solid #FF0000; text-align: left;"> $$\displaystyle <P_n,1>_w $$ = 0, Why this equality holds? Hylon.Chen 20:45, 24 January 2012 (UTC)

since $$\displaystyle P_0 (x) = 1$$, and by the orthogonality of the family of Legendre polynomials $$\displaystyle \{ P_n \}$$, you have $$\displaystyle <P_n, P_0> = <P_n,1>_w = 0 , \text{ for } n \ge 1$$. Egm6322.s12 15:38, 26 January 2012 (UTC)

For many orthogonal polynomials, their zeros are non-uniformly distributed, while clustered towards the endpoints of the interval $$\displaystyle [-1,1]$$.

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$$\displaystyle I[f] := \int_{I}f(x)w(x)dx $$ (ab40) and define a integration formula with $$\displaystyle n \geq 1$$ points,
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$$\displaystyle U^n[f] := \sum_{i = 1}^{n}f(x^i)w^i $$ (ab41) where $$\displaystyle x^i $$ are a set of nodes and $$\displaystyle w^i $$ are integration weights. We need to find a set of $$\displaystyle \{ x^i,w^i\} $$ such that $$\displaystyle U^n [f] \approx I [f]$$, and hopefully $$\displaystyle \lim_{n\rightarrow \infty}U^n[f] = I [f]$$. Highly accurate integration formulas can be constructed using orthogonal polynomials. Let $$\displaystyle P_n, \;\;n \in \mathbb N$$ be orthogonal polynomials, and let $$\displaystyle z_k^N, \;\;k = 1,2,3,...,N$$ be the zeros of $$\displaystyle P_n $$. Let $$\displaystyle L_k^{N-1}$$ be the $$\displaystyle N-1 $$th-degree Lagrange polynomials through the nodes $$\displaystyle z_k^N$$, and let $$\displaystyle \Pi_{N-1}f(x)=\sum_{k = 1}^{N}f(z_k^N)L_k^{N-1}$$ be the $$\displaystyle N-1$$th-degree interpolation of $$\displaystyle f(x)$$. Then the integration in (ab40) can be approximated by integrating $$\displaystyle \Pi_{N-1}f(x)$$.
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$$\displaystyle \int_{I}f(x)w(x)dx = \sum_{k = 1}^{N}f(z_k^N)w_k $$ (ab42) where
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$$\displaystyle w_k = \int_{I}L_k^{N-1}w(x)dx\,\,\,\, 1\leq k \leq N $$ (ab43) are the weights.
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Formula (ab42) become exact when $$\displaystyle f(x)$$ is any polynomial with degree less than or equal to $$\displaystyle 2N - 1$$ in $$\displaystyle I $$.

See Gaussian quadratureWiki[[media:nm1.s11.mtg7.djvu|NM1S11 - Mtg 7]], Clenshaw-Curtis quadrature(Wiki) and Newton-Cotes quadrature rulesWiki[[media:nm1.s11.mtg8.djvu|NM1S11 - Mtg 8]].

=References=