Secondary Measure - Case of The Lebesgue Measure and Some Other Examples

Case of The Lebesgue Measure and Some Other Examples

The Lebesgue measure on the standard interval is obtained by taking the constant density ρ(x) = 1.

The associated orthogonal polynomials are called Legendre polynomials and can be clarified by

The norm of P_n is worth

The recurrence relation in three terms is written:

The reducer of this measure of Lebesgue is given by

The associated secondary measure is then clarified as

.

If we normalize the polynomials of Legendre, the coefficients of Fourier of the reducer φ related to this orthonormal system are null for an even index and are given by

for an odd index n.

The Laguerre polynomials are linked to the density ρ(x) = e−x on the interval I = [0, ∞). They are clarified by

and are normalized.

The reducer associated is defined by

The coefficients of Fourier of the reducer φ related to the Laguerre polynomials are given by

This coefficient C_n(φ) is no other than the opposite of the sum of the elements of the line of index n in the table of the harmonic triangular numbers of Leibniz.

The Hermite polynomials are linked to the Gaussian density

on I = R.

They are clarified by

and are normalized.

The reducer associated is defined by

The coefficients of Fourier of the reducer φ related to the system of Hermite polynomials are null for an even index and are given by

for an odd index n.

The Chebyshev measure of the second form. This is defined by the density

on the interval .

It is the only one which coincides with its secondary measure normalised on this standard interval. Under certain conditions it occurs as the limit of the sequence of normalized secondary measures of a given density.