Secondary Measure
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 that we will specify further, it is possible to obtain the existence of a secondary measure and even to express it. For example, if one works in the Hilbert space ''L''2(, 1 R, ρ) : \forall x \in ,1 \qquad \mu(x)=\frac with : \varphi(x) = \lim_ 2\int_0^1\frac \, dt in the general case, or: : \varphi(x) = 2\rho(x)\text\left(\frac\right) - 2 \int_0^1\frac \, 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_(z)=z-c_1-\frac in which ''c''1 is the moment of order 1 of the measure ρ. These secondary measures, and the theory around th ...
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Measure (mathematics)
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Co ...
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