In mathematics, the secondary measure associated with a measure of positive density ρ when there is one, is a measure of positive density μ, turning the secondary polynomials associated with the orthogonal polynomials for ρ into an orthogonal system.
Introduction Under certain assumptions, it is possible to obtain the existence of a secondary measure and even to express it. For example, this can be done when working in the Hilbert space L2([0, 1], R, ρ)
∀ x ∈ [ 0 , 1 ] , μ ( x ) = ρ ( x ) φ 2 ( x ) 4 + π 2 ρ 2 ( x ) {\displaystyle \forall x\in [0,1],\qquad \mu (x)={\frac {\rho (x)}{{\frac {\varphi ^{2}(x)}{4}}+\pi ^{2}\rho ^{2}(x)}}}
with
φ ( x ) = lim ε → 0 + 2 ∫ 0 1 ( x − t ) ρ ( t ) ( x − t ) 2 + ε 2 d t {\displaystyle \varphi (x)=\lim _{\varepsilon \to 0^{+}}2\int _{0}^{1}{\frac {(x-t)\rho (t)}{(x-t)^{2}+\varepsilon ^{2}}}\,dt}
in the general case, or:
φ ( x ) = 2 ρ ( x ) ln ( x 1 − x ) − 2 ∫ 0 1 ρ ( t ) − ρ ( x ) t − x d t {\displaystyle \varphi (x)=2\rho (x){\text{ln}}\left({\frac {x}{1-x}}\right)-2\int _{0}^{1}{\frac {\rho (t)-\rho (x)}{t-x}}\,dt}
when ρ satisfies a Lipschitz condition. This application φ is called the reducer of ρ. More generally, μ et ρ are linked by their Stieltjes transformation with the following formula:
S μ ( z ) = z − c 1 − 1 S ρ ( z ) {\displaystyle S_{\mu }(z)=z-c_{1}-{\frac {1}{S_{\rho }(z)}}}
in which c1 is the moment of order 1 of the measure ρ. Secondary measures and the theory around them may be used to derive traditional formulas of analysis concerning the Gamma function, the Riemann zeta function, and the Euler–Mascheroni constant. They have also allowed the clarification of various integrals and series, although this tends to be difficult a priori. Finally they make it possible to solve integral equations of the form
f ( x ) = ∫ 0 1 g ( t ) − g ( x ) t − x ρ ( t ) d t {\displaystyle f(x)=\int _{0}^{1}{\frac {g(t)-g(x)}{t-x}}\rho (t)\,dt}
where g is the unknown function, and lead to theorems of convergence towards the Chebyshev and Dirac measures.
The broad outlines of the theory Let ρ be a measure of positive density on an interval I and admitting moments of any order. From this, a family {Pn} of orthogonal polynomials for the inner product induced by ρ can be created. Let {Qn} be the sequence of the secondary polynomials associated with the family P. Under certain conditions there is a measure for which the family Q is orthogonal. This measure, which can be clarified from ρ, is called a secondary measure associated initial measure ρ. When ρ is a probability density function, a sufficient condition that allows μ to be a secondary measure associated with ρ while admitting moments of any order is that its Stieltjes Transformation is given by an equality of the type
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