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Stieltjes Transformation
In mathematics, the Stieltjes transformation of a measure of density on a real interval is the function of the complex variable defined outside by the formula S_(z)=\int_I\frac, \qquad z \in \mathbb \setminus I. Under certain conditions we can reconstitute the density function starting from its Stieltjes transformation thanks to the inverse formula of Stieltjes-Perron. For example, if the density is continuous throughout , one will have inside this interval \rho(x)=\lim_ \frac. Connections with moments of measures If the measure of density has moments of any order defined for each integer by the equality m_=\int_I t^n\,\rho(t)\,dt, then the Stieltjes transformation of admits for each integer the asymptotic expansion in the neighbourhood of infinity given by S_(z)=\sum_^\frac+o\left(\frac\right). Under certain conditions the complete expansion as a Laurent series can be obtained: S_(z) = \sum_^\frac. Relationships to orthogonal polynomials The correspondence (f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Secondary Polynomials
In mathematics, the secondary polynomials \ associated with a sequence \ of polynomials orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ... with respect to a density \rho(x) are defined by : q_n(x) = \int_\mathbb\! \frac \rho(t)\,dt. To see that the functions q_n(x) are indeed polynomials, consider the simple example of p_0(x)=x^3. Then, :\begin q_0(x) & = \int_\mathbb \! \frac \rho(t)\,dt \\ & = \int_\mathbb \! \frac \rho(t)\,dt \\ & = \int_\mathbb \! (t^2+tx+x^2)\rho(t)\,dt \\ & = \int_\mathbb \! t^2\rho(t)\,dt + x\int_\mathbb \! t\rho(t)\,dt + x^2\int_\mathbb \! \rho(t)\,dt \end which is a polynomial x provided that the three integrals in t (the moments of the density \rho) are convergent. See also * Secondary measure Polynomials {{algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonality, orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special cases. The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by Pafnuty Chebyshev, P. L. Chebyshev and was pursued by Andrey Markov, A. A. Markov and Thomas Joannes Stieltjes, T. J. Stieltjes. They appear in a wide variety of fields: numerical analysis (Gaussian quadrature, quadrature rules), probability theory, representation theory (of Lie group, Lie groups, quantum group, quantum groups, and re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convergent (continued Fraction)
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers a_i are called the coefficients or terms of the continued fraction. It is generally assumed that the numerator of all of the fractions is 1. If arbitrary values and/or functions are used in place of one or more of the numerators or the integers in the denominators, the resulting expression is a generalized continued fraction. When it is necessa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Continued Fraction
In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary complex values. A generalized continued fraction is an expression of the form :x = b_0 + \cfrac where the () are the partial numerators, the are the partial denominators, and the leading term is called the ''integer'' part of the continued fraction. The successive convergents of the continued fraction are formed by applying the fundamental recurrence formulas: :\begin x_0 &= \frac = b_0, \\ pxx_1 &= \frac = \frac, \\ pxx_2 &= \frac = \frac,\ \dots \end where is the ''numerator'' and is the ''denominator'', called continuants, of the th convergent. They are given by the recursion :\begin A_n &= b_n A_ + a_n A_, \\ B_n &= b_n B_ + a_n B_ \qquad \text n \ge 1 \end with initial values :\begin A_ &= 1,& A_0&=b_0,\\ B_&=0, & B_0&=1. \end If the sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Secondary Polynomials
In mathematics, the secondary polynomials \ associated with a sequence \ of polynomials orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ... with respect to a density \rho(x) are defined by : q_n(x) = \int_\mathbb\! \frac \rho(t)\,dt. To see that the functions q_n(x) are indeed polynomials, consider the simple example of p_0(x)=x^3. Then, :\begin q_0(x) & = \int_\mathbb \! \frac \rho(t)\,dt \\ & = \int_\mathbb \! \frac \rho(t)\,dt \\ & = \int_\mathbb \! (t^2+tx+x^2)\rho(t)\,dt \\ & = \int_\mathbb \! t^2\rho(t)\,dt + x\int_\mathbb \! t\rho(t)\,dt + x^2\int_\mathbb \! \rho(t)\,dt \end which is a polynomial x provided that the three integrals in t (the moments of the density \rho) are convergent. See also * Secondary measure Polynomials {{algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moment (mathematics)
In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total mass) is the center of mass, and the second moment is the moment of inertia. If the function is a probability distribution, then the first moment is the expected value, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. The mathematical concept is closely related to the concept of moment in physics. For a distribution of mass or probability on a bounded interval, the collection of all the moments (of all orders, from to ) uniquely determines the distribution (Hausdorff moment problem). The same is not true on unbounded intervals (Hamburger moment problem). In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think systematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonality, orthogonal to each other under some inner product. The most widely used orthogonal polynomials are the classical orthogonal polynomials, consisting of the Hermite polynomials, the Laguerre polynomials and the Jacobi polynomials. The Gegenbauer polynomials form the most important class of Jacobi polynomials; they include the Chebyshev polynomials, and the Legendre polynomials as special cases. The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by Pafnuty Chebyshev, P. L. Chebyshev and was pursued by Andrey Markov, A. A. Markov and Thomas Joannes Stieltjes, T. J. Stieltjes. They appear in a wide variety of fields: numerical analysis (Gaussian quadrature, quadrature rules), probability theory, representation theory (of Lie group, Lie groups, quantum group, quantum groups, and re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
