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Szegő Polynomial
In mathematics, a Szegő polynomial is one of a family of orthogonal polynomials for the Hermitian inner product :\langle f, g\rangle = \int_^f(e^)\overline\,d\mu where dμ is a given positive measure on minus;π, π Writing \phi_n(z) for the polynomials, they obey a recurrence relation :\phi_(z)=z\phi_n(z) + \rho_\phi_n^*(z) where \rho_ is a parameter, called the ''reflection coefficient'' or the ''Szegő parameter''. See also * Cayley transform * Schur class * Favard's theorem In mathematics, Favard's theorem, also called the Shohat–Favard theorem, states that a sequence of polynomials satisfying a suitable three-term recurrence relation is a sequence of orthogonal polynomials. The theorem was introduced in the theor ... References * * G. Szegő, "Orthogonal polynomials", Colloq. Publ., 33, Amer. Math. Soc. (1967) Orthogonal polynomials {{polynomial-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 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] |
Cayley Transform
In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. The transform is a homography used in real analysis, complex analysis, and quaternionic analysis. In the theory of Hilbert spaces, the Cayley transform is a mapping between linear operators . Real homography A simple example of a Cayley transform can be done on the real projective line. The Cayley transform here will permute the elements of in sequence. For example, it maps the positive real numbers to the interval ��1, 1 Thus the Cayley transform is used to adapt Legendre polynomials for use with functions on the positive real numbers with Legendre rational functions. As a real homography, points are described with projective coordinates, and the mapping is : ,\ 1= \left frac ,\ 1\right\thicksim - 1, \ x + 1= ,\ 1begin1 & 1 \\ -1 & 1 \end ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur Class
In complex analysis, the Schur class is the set of holomorphic functions f(z) defined on the open unit disk \mathbb = \ and satisfying , f(z), \leq 1 that solve the Schur problem: Given complex numbers c_0,c_1,\dotsc,c_n, find a function :f(z) = \sum_^ c_j z^j + \sum_^f_j z^j which is analytic and bounded by on the unit disk. The method of solving this problem as well as similar problems (e.g. solving Toeplitz systems and Nevanlinna-Pick interpolation) is known as the Schur algorithm (also called Coefficient stripping or Layer stripping). One of the algorithm's most important properties is that it generates orthogonal polynomials which can be used as orthonormal basis functions to expand any th-order polynomial. It is closely related to the Levinson algorithm though Schur algorithm is numerically more stable and better suited to parallel processing. Schur function Consider the Carathéodory function of a unique probability measure d\mu on the unit circle \mathbb =\ giv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Favard's Theorem
In mathematics, Favard's theorem, also called the Shohat–Favard theorem, states that a sequence of polynomials satisfying a suitable three-term recurrence relation is a sequence of orthogonal polynomials. The theorem was introduced in the theory of orthogonal polynomials by and , though essentially the same theorem was used by Stieltjes in the theory of continued fraction A continued fraction is a mathematical expression that can be written as a fraction with a denominator that is a sum that contains another simple or continued fraction. Depending on whether this iteration terminates with a simple fraction or not, ...s many years before Favard's paper, and was rediscovered several times by other authors before Favard's work. Statement Suppose that ''y''0 = 1, ''y''1, ... is a sequence of polynomials where ''y''''n'' has degree ''n''. If this is a sequence of orthogonal polynomials for some positive weight function then it satisfies a 3-term recurrence relation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |