Secondary Measure - A Few Beautiful Applications

A Few Beautiful Applications

In the formulas below G is Catalan's constant, γ is the Euler's constant, β_2n is the Bernoulli number of order 2n, H_2n+1 is the harmonic number of order 2n+1 and Ei is the Exponential integral function.

The notation indicating the 2 periodic function coinciding with on (−1, 1).

If the measure ρ is reducible and let φ be the associated reducer, one has the equality

If the measure ρ is reducible with μ the associated reducer, then if f is square integrable for μ, and if g is square integrable for ρ and is orthogonal with P₀ = 1 one has equivalence:

c₁ indicates the moment of order 1 of ρ and T_ρ the operator