Orthogonal polynomials

Introduction

In Mathematical, a continuation of orthogonal polynomials is a infinite continuation of Polynôme S p ₀ ( X ), p ₁ ( X ), p ₂ ( X )…, in which each p _N ( X ) has a degree N and so that the polynomials of the continuation are orthogonal two to two, with the following direction:

We can define the scalar Produit functions, (by analogy with the scalar product for the Vecteur S), in integral the product of these functions:

$\backslash \; langle\; F,\; G\; \backslash \; rangle=\; \backslash \; int\_\; \{x\_1\}\; ^\; \{x\_2\}\; F\; (X)\; G\; (X)\; \backslash ,\; dx$

more generally, we can introduce a " function poids" W ( X ) in the integral:

$\backslash \; langle\; F,\; G\; \backslash \; rangle=\; \backslash \; int\_\; \{x\_1\}\; ^\; \{x\_2\}\; F\; (X)\; G\; (X)\; W\; (X)\; \backslash ,\; dx$

With this definition of scalar product, two functions are orthogonal between them if their scalar product is equal to zero (same manner that two vectors are orthogonal (perpendiculars) if their scalar product equalizes zero).

Such a made scalar product of the whole of all the functions of standard finished, a Space of Hilbert.

The interval of integration is called interval of orthogonality . It can be infinite on one or two terminals.

This field of the orthogonal polynomials was developed during the XIXe century by the study of the continued fractions by Stieltjes. In rose from multiple applications in mathematics and Physique.

Example: polynomials of Legendre

The orthogonal polynomials simplest are the Polynômes of Legendre for which the interval of orthogonality is and the function weight is simply the constant function of value 1:

$P\_0\; (X)\; =\; 1\; \backslash ,$

$P\_1\; (X)\; =\; X\; \backslash ,$

$P\_2\; (X)\; =\; \backslash \; frac\; \{3x^2-1\}\; \{2\}\; \backslash ,$

$P\_3\; (X)\; =\; \backslash \; frac\; \{5x^3-3x\}\; \{2\}\; \backslash ,$

$P\_4\; (X)\; =\; \backslash \; frac\; \{35x^4-30x^2+3\}\; \{8\}\; \backslash ,$

$\backslash \; dowries\; \backslash ,$

They all are orthogonal on:

$\backslash \; int\_\; \{-\; 1\}\; ^\; \{1\}\; P\_m\; (X)\; P\_n\; (X)\; \backslash ,\; dx\; =\; 0\; \backslash \; qquad\; \backslash \; mathrm\; \{for\}\; \backslash \; qquad\; m\; \backslash \; n$

The function weight must be strictly positive in the field of integration. In certain cases, it can be null or infinite at the boundaries of the integral. The integral of the product of the function weight by a polynomial must be finished.

Any continuation of polynomials $p\_0,\; p\_1\; \backslash \; dots$, where each $\backslash \; p\_k$ is of degree K , is a base of the vector space (of infinite size) of all the polynomials. A succession of orthogonal polynomials is simply a continuation which includes/understands an orthogonal base for this space, relative with this scalar product.

The Procédé of Gram-Schmidt can transform any base of a vector space (provided with a scalar product) into an orthogonal base. One starts with a vector and while incorporating, one by one, of new vectors in such a way that each new vector is orthogonal with all the precedents. This is carried out by withdrawing a linear Combinaison preceding vectors. This is often a exercise for the first elementary courses of Linear algebra. It results the polynomials from them from Legendre.

When an orthogonal base is built, one can be tried to make it orthonormal , i.e., in which $\backslash \; langle\; p\_n,\; p\_n\; \backslash \; rangle\; \backslash \; =\; \backslash \; 1$. In the case of the polynomials, it often results from it from dreadful square roots for the coefficients. In the place, the polynomials are transformed, in a way approved by the mathematicians, so that the coefficients give simpler formulas. One calls that standardization . The polynomials " classiques" enumerated below were standardized. Typically, their coefficients of the term moreover high degree were put at a given quantity. This standardization does not have mathematical significance, it is right a convention. Standardization implies also a scaling of the function weight.

Once the continuation of polynomials standardized, one can define the standard. That is to say

$h\_n=\; \backslash \; langle\; p\_n,\; \backslash \; p\_n\; \backslash \; rangle$

The standard is the square root of this. The values of $\backslash \; h\_n$ for standardization are enumerated in the table below. We have

$\backslash \; langle\; p\_m,\; \backslash \; p\_n\; \backslash \; rangle\; \backslash \; =\; \backslash \; \backslash \; delta\_\; \{mn\}\; h\_n$

where δ_mn is the Delta of Kronecker.

Properties of the continuations of orthogonal polynomials

Any continuation of orthogonal polynomials has a great number of elegant and attractive properties. To start:

Lemma 1: Being taken an action pursuant of orthogonal polynomials $\backslash \; p\_i\; (X)$, any polynomial $\backslash \; S\; (X)$ of degree N can be expressed in a single way like a linear combination of $p\_0\; \backslash \; dowries\; p\_n$. I.e., there exist coefficients $\{\backslash \; alpha\}\; \_0\; \backslash \; dowries\; \{\backslash \; alpha\}\; \_n$ such as

$S\; (X)\; =\; \backslash \; sum\_\; \{i=0\}\; ^n\; \{\backslash \; alpha\}\; \_i\; \backslash \; p\_i\; (X)$

Lemma 2: Being given a succession of orthogonal polynomials, any element of this continuation is orthogonal with any polynomial of strictly lower degree.

Relation of recurrence

For any continuation of orthogonal polynomials, there relative exists a relation of Récurrence to three consecutive polynomials.

$p\_\; \{n+1\}\; \backslash \; =\; \backslash \; (a\_nx+b\_n)\; \backslash \; p\_n\; \backslash \; -\; \backslash \; c\_n\; \backslash \; p\_\; \{n-1\}$

The coefficients has , B , and C depends on N . They also depend on the standardization, obviously.

The values of $a\_n$, $b\_n$ and $c\_n$ can be calculated directly. Are $k\_j$ and $k\_j\text{'}$ the first two coefficients of $p\_j$:

$p\_j\; (X)\; =k\_jx^j+k\_j\text{'}\; x^\; \{j-1\}\; +\; \backslash \; cdots$

and $h\_j$ the scalar product of $p\_j$ by itself:

$h\_j\; \backslash \; =\; \backslash \; \backslash \; langle\; p\_j,\; \backslash \; p\_j\; \backslash \; rangle$

One obtains

$a\_n=\; \backslash \; frac\; \{k\_\; \{n+1\}\}\; \{k\_n\}\; \backslash \; \backslash \; \backslash \; \backslash \; \backslash \; \backslash \; \backslash \; b\_n=a\_n\; \backslash \; left\; (\backslash \; frac\; \{k\_\; \{n+1\}\; \text{'}\}\; \{k\_\; \{n+1\}\}\; -$

\ frac {k_n'} {k_n} \ right) \ \ \ \ \ \ \ c_n=a_n \ left (\ frac {k_ {n-1} h_n} {k_n h_ {n-1}} \ right).

Existence of real roots

Any polynomial of a succession of orthogonal polynomials of which degree N is equal to or higher than 1 admits N distinct roots, all real, and located strictly inside the interval of integration.

(Whoever already drew the curve representative of a polynomial, knows at which point it is rare, for a polynomial whose coefficients were randomly selected, to have all its real roots.)

Position of the roots

The roots of the polynomials are strictly between the roots of the polynomial of higher degree in the continuation.

Differential equations leading to orthogonal polynomials

An important class of the orthogonal polynomials comes from a differential equation of the form

$\{Q\; (X)\}\backslash ,\; F$ + {L (X)}\, f' + {\ lambda} F = 0 \,

where Q is a given quadratic polynomial and L a given linear polynomial. The function F and the constant λ are the unknown factors.

(To notice that a polynomial solution has all its direction for such an equation.

Each term of the equation is a polynomial, and the degrees are in conformity).

The solutions of this differential equation have singularities, unless λ does not take specific values. The continuation of numbers $\{\backslash \; lambda\}\; \_0,\; \{\backslash \; lambda\}\; \_1,\; \{\backslash \; lambda\}\; \_2\; \backslash \; dowries\; \backslash ,$ leads to a succession of polynomials solutions $P\_0,\; P\_1,\; P\_2\; \backslash \; dowries\; \backslash ,$ if one of the following assertions is checked:

1. Q is really quadratic, L is linear, Q has two distinct real roots, the root of L is located between the two roots of Q, and the terms moreover high degree of Q and L have the same sign.

2. Q is not quadratic, but linear, L is linear, the roots of Q and L are different, and the terms moreover high degree of Q and L have the same sign if the root of L is smaller than that of Q, or conversely.
3. Q is a polynomial constant not no one, L is linear, and the term moreover high degree of L is of sign opposed to that of Q.

These three cases lead respectively to the polynomials of Jacobi , Laguerre and Hermite . For each one of these cases:

• the solution is a succession of polynomials $P\_0,\; P\_1,\; P\_2\; \backslash \; dowries\; \backslash ,$, each $P\_n\; \backslash ,$ having a degree N, and corresponding to the number $\{\backslash \; lambda\}\; \_n\; \backslash ,$.

• the interval of orthogonality is limited by the roots of Q.
• the root of L is inside the interval of orthogonality.
• By noting $R\; (X)\; =\; e^\; \{\backslash \; int\; \backslash \; frac\; \{L\; (X)\}\{Q\; (X)\}\backslash ,\; dx\}\; \backslash ,$, the polynomials are orthogonal under the function weight $W\; (X)\; =\; \backslash \; frac\; \{R\; (X)\}\{Q\; (X)\}\backslash ,$
• W (X) cannot cancel or take an infinite value in the interval, although it can do it on the extremities.
• W (X) can be selected to be positive on the interval (to multiply the differential equation by -1 if necessary)

Because of the constant of integration, the quantity R (X) is defined except for a multiplicative constant. The table low gives the values " officielles" R (X) and W (X).

Formulate of Rodrigues

With the assumptions of the preceding section, P _N ( X ) is proportional to $\backslash \; frac\; \{1\}\; \{W\; (X)\}\; \backslash \; \backslash \; frac\; \{d^n\}\; \{dx^n\}\; \backslash \; left\; (W\; (X)\; ^n\; \backslash \; right)$

better known equation under the name of “formula of Rodrigues”. She is often written:

$P\_n\; (X)\; =\; \backslash \; frac\; \{1\}\; \backslash \; \backslash \; frac\; \{d^n\}\; \{dx^n\}\; \backslash \; left\; (W\; (X)\; ^n\; \backslash \; right)$

where the numbers E _N depend on standardization. The values of E _N are given in the table low.

The λ_n numbers

With the assumptions of the preceding section,

$\{\backslash \; lambda\}\; \_n\; =\; -\; N\; \backslash \; left\; (\backslash \; frac\; \{n-1\}\; \{2\}\; \backslash \; Q$ + the \ right)

(Since $Q$ is quadratic and $L$ linear, $Q$ and $L\text{'}$ is constant, they are indeed numbers.)

Second form of the differential equation

With $R\; (X)\; =\; e^\; \{\backslash \; int\; \backslash \; frac\; \{L\; (X)\}\{Q\; (X)\}\backslash ,\; dx\}\; \backslash ,$.

Then

$(Ry\text{'})\; “=\; R\; \backslash ,\; there$ + R” \, y' = R \, there + \ frac {R \, L} {Q} \, y'

By multiplying the differential equation now

$\{Q\}\; \backslash ,\; there$ + {L} \, y' + {\ lambda} \, there = 0 \,

by R/Q, one obtains

$R\; \backslash ,\; there$ + \ frac {R \, L} {Q} \, y' + \ frac {R \, \ lambda} {Q} \, there = 0 \,

or

$(Ry\text{'})\; \text{'}+\; \backslash \; frac\; \{R\; \backslash ,\; \backslash \; lambda\}\; \{Q\}\; \backslash ,\; there\; =\; 0\; \backslash ,$

It is forms it standardized Sturm-Liouville of the equation.

Third form of the differential equation

By posing $S\; (X)\; =\; \backslash \; sqrt\; \{R\; (X)\}\; =\; e^\; \{\backslash \; int\; \backslash \; frac\; \{L\; (X)\}\{2\; \backslash ,\; Q\; (X)\}\backslash ,\; dx\}\; \backslash ,$.

Then:

$S\text{'}\; =\; \backslash \; frac\; \{S\; \backslash ,\; L\}\; \{2\; \backslash ,\; Q\}.$

By multiplying the differential equation now

$\{Q\}\; \backslash ,\; there$ + {L} \, y' + {\ lambda} \, there = 0 \,

by S/Q, one obtains:

$S\; \backslash ,\; there$ + \ frac {S \, L} {Q} \, y' + \ frac {S \, \ lambda} {Q} \, there = 0 \,

or

$S\; \backslash ,\; there$ + 2 \, \, y' + \ frac {S \, \ lambda} {Q} \, there = 0 \,

But $(S\; \backslash ,\; there)$ = S \, there + 2 \, \, y' + S \, y, therefore

$(S\; \backslash ,\; there)$ + \ left (\ frac {S \, \ lambda} {Q} - S \ right) \, there = 0, \,

or, by posing U = Sy ,

$u$ + \ left (\ frac {\ lambda} {Q} - \ frac {S } {S} \ right) \, U = 0. \,

Table of the traditional orthogonal polynomials

See too

• random Matrix
• secondary Polynomials
• secondary Measurements

References

• Polynomials of Tchebychev on Maths-Linux .

Orthogonal

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