|
Secondary Polynomials
In mathematics, the secondary polynomials \ associated with a sequence \ of polynomials orthogonal 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 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 ... Polynomials References [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
|
 |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |