an introduction
Legendre . polynomial
The Legendre . polynomial are solutions of the Legendre differential equation, and form the simplest example of a sequence orthogonal polynomials (In mathematics, a sequence of orthogonal polynomials is an infinite sequence of…).
Legendre equation
We call the Legendre equation:
Thus we define the Polynomial Legendre (Legendre polynomials are solutions of the differential equation for…) s_{Not} (for everything (All is understood as all that exists most often as the world or …) all natural (In mathematics, a natural number is a positive number (or zero) that basically allows…) Not):
so we have where You know? polynomial (A polynomial, in mathematics, is the linear summation of the products of…) Jacobi from index Not associated with the parameters α and .
Some polynomials
The first polynomials are:
Other definitions
bonnet repetition formula
and for everyone Not> 0
Formula Rodriguez (Rodrigues is the smallest of the three islands in the Mascarene Archipelago.)
Define the polynomial s_{Not} (for all natural integers Not) by :
identification (Definition is a letter that says what the thing is or what the name means. Hence…) Analytical
We can also define this sequence of polynomials by its generating function:
The theory (A theory is a proposition that can be demonstrated mathematically, i.e….) Then give the residue:
where is the Outline (COMET Nucleus TOUR (CONTOUR) is a NASA space probe that is part of the program…) Surrounds the original and gets caught in the meaning (SENS (Neglected Engineering Aging Strategies) is a scientific project aimed at…) Furniture Calculation.
Tariffs as a total
We define this polynomial in two ways in the form of a sum:
(infer )
Sequential decomposition of Legendre polynomials
Holomorphic function decomposition
Any function f, holomorphic inside an ellipse of foci 1 and +1, can be written as a series that converges uniformly inside the ellipse:
with
Decomposition of the Lipschitz function
note Polynomial quotient s_{Not} by it Basic (Standard, from the Latin norma (“square, ruler”) designating…).
This is it F Continuous application on [1,1]. Everything is natural Not is put
Then who is he Field (A square is a regular foursided polygon. This means that…) In short, and makes it possible to explain the orthogonal projection of F on me :
We also have:
 with
Further suppose that f is a Lipschitz function. Then we have the additional property:^{[réf. souhaitée]}
autrement dit, l’égalité
est vraie non seulement au sens L^{2} mais au sens de la convergence simple (La convergence simple ou ponctuelle est un critère de convergence dans un espace fonctionnel,…) sur ]1.1[.
Propriétés
Degré (Le mot degré a plusieurs significations, il est notamment employé dans les domaines…)
Le polynôme P_{n} est de degré n.
Parité
Les polynômes de Legendre suivent la parité de n. On peut exprimer cette propriété par :
(en particulier, P_{n}( − 1) = ( − 1)^{n} et P_{2n + 1}(0) = 0).
Orthogonalité (En mathématiques, l’orthogonalité est un concept d’algèbre linéaire…)
Les polynômes orthogonaux les plus simples sont les polynômes de Legendre pour lesquels l’intervalle d’orthogonalité est [−1, 1] and the job weight (Weight is the gravitational force, the gravitational force and the origin of inertia, which it exerts…) is simply a constant function of the value 1: these polynomials are orthogonal with respect to scalar product (In vector geometry, the point product is an algebraic operation…) set to in relation:

identification s_{Not} indicates that clean vector (In mathematics, the eigenvector concept is an algebraic concept that applies to…) to special value (In mathematics, the eigenvector concept is an algebraic concept that applies to…) n(n+1) to resemble:
But this similarity is the same with the product scalar (The real scalar is a number independent of the choice of base chosen to express…) Above, because integration by parts shows this
 :Basic
base square, in The^{2}([1,1]), he is
In fact, for all n > 1, we can establish the relationship
From which we deduce (use that for everyone KAnd the s“_{K −1} class K2k orthogonal to that s_{K}and implementing integration by parts):
 s_{Not}s_{Not + 1} It’s weird for everyone KAnd the s_{K}(1) = 1thus we arrive at (2Not + 1)   s_{Not}   ^{2} = 2.
 s_{Not}s_{Not + 1} It’s weird for everyone KAnd the s_{K}(1) = 1thus we arrive at (2Not + 1)   s_{Not}   ^{2} = 2.
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