- - - figure the first normalized legendre polynomials as the legendre polynomials form an orthonormal basis of ( {- dx)they can be used to approximate arbitrary functions in this space for examplefigure shows an approximation using the first legendre polynomials ( of the fourier expansion of the indicator function on the interval (- / / these legendre polynomials form the basis of -dimensional linear subspace onto which the indicator function is orthogonally projected - - figure approximation of the indicator function on the interval (- / / )using the legendre polynomials orthogonal polynomials laguerre polynomials the legendre polynomials were produced in the following waywe started with an unnormalized probability density on -in this case the probability density of the uniform distribution on (- we then constructed sequence of polynomials by applying the gram-schmidt procedure to the monomials xx by using exactly the same procedurebut with different probability densitywe can produce other such orthogonal polynomials for examplethe density of the standard exponential distributionw(xe- gives the laguerre polynomialswhich are defined this can be further generalized to the density of gamma distribution