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Biorthogonal Polynomials
In mathematics, a biorthogonal polynomial is a polynomial that is orthogonal to several different measures. Biorthogonal polynomials are a generalization of orthogonal polynomials and share many of their properties. There are two different concepts of biorthogonal polynomials in the literature: introduced the concept of polynomials biorthogonal with respect to a sequence of measures, while Szegő introduced the concept of two sequences of polynomials that are biorthogonal with respect to each other. Polynomials biorthogonal with respect to a sequence of measures A polynomial ''p'' is called biorthogonal with respect to a sequence of measures ''μ''1, ''μ''2, ... if :\int p(x) \, d\mu_i(x) =0 whenever ''i'' ≤ deg(''p''). Biorthogonal pairs of sequences Two sequences ''ψ''0, ''ψ''1, ... and ''φ''0, ''φ''1, ... of polynomials are called biorthogonal (for some measure ''μ'') if : \int \phi_m(x)\psi_n(x) \, d\mu(x) = 0 whenever ''m'' ≠ ''n''. The definition of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Orthogonal Polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ... 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. These are frequently given by the Rodrigues' formula. The field of orthogonal polynomials developed in the late 19th century from a study of continued fractions by Pafnuty Chebyshev, P. L. Chebyshev and wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Transactions Of The American Mathematical Society
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of pure and applied mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 printed pages. Its ISSN number is 0002-9947. See also * ''Bulletin of the American Mathematical Society'' * ''Journal of the American Mathematical Society'' * '' Memoirs of the American Mathematical Society'' * '' Notices of the American Mathematical Society'' * ''Proceedings of the American Mathematical Society'' References External links * ''Transactions of the American Mathematical Society''on JSTOR JSTOR ( ; short for ''Journal Storage'') is a digital library of academic journals, books, and primary sources founded in 1994. Originally containing digitized back issues of academic journals, it now encompasses books and other primary source ... American Mathematical Society academic journals Mathematics jo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |