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Schur Parameters
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 =\ gi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear engineering, nuclear, aerospace engineering, aerospace, mechanical engineering, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to its Taylor series (that is, it is Analyticity of holomorphic functions, analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions). History Complex analysis is one of the classical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holomorphic Function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivative in a neighbourhood is a very strong condition: it implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (''analytic''). Holomorphic functions are the central objects of study in complex analysis. Though the term ''analytic function'' is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighbourhood of each point in its domain. That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis. Holomorphic functions are also sometimes referred to as ''regular fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toeplitz Matrix
In linear algebra, a Toeplitz matrix or diagonal-constant matrix, named after Otto Toeplitz, is a matrix in which each descending diagonal from left to right is constant. For instance, the following matrix is a Toeplitz matrix: :\qquad\begin a & b & c & d & e \\ f & a & b & c & d \\ g & f & a & b & c \\ h & g & f & a & b \\ i & h & g & f & a \end. Any ''n'' × ''n'' matrix ''A'' of the form :A = \begin a_0 & a_ & a_ & \cdots & \cdots & a_ \\ a_1 & a_0 & a_ & \ddots & & \vdots \\ a_2 & a_1 & \ddots & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & a_ & a_ \\ \vdots & & \ddots & a_1 & a_0 & a_ \\ a_ & \cdots & \cdots & a_2 & a_1 & a_0 \end is a Toeplitz matrix. If the ''i'', ''j'' element of ''A'' is denoted ''A''''i'', ''j'' then we have :A_ = A_ = a_. A Toeplitz matrix is not necessarily square. Solving a Toeplitz system A matrix equation of the form :Ax = b is called a Toeplitz system ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nevanlinna–Pick Interpolation
In complex analysis, given ''initial data'' consisting of n points \lambda_1, \ldots, \lambda_n in the complex unit disc \mathbb and ''target data'' consisting of n points z_1, \ldots, z_n in \mathbb, the Nevanlinna–Pick interpolation problem is to find a holomorphic function \varphi that interpolates the data, that is for all i \in \, :\varphi(\lambda_i) = z_i, subject to the constraint \left\vert \varphi(\lambda) \right\vert \le 1 for all \lambda \in \mathbb. Georg Pick and Rolf Nevanlinna solved the problem independently in 1916 and 1919 respectively, showing that an interpolating function exists if and only if a matrix defined in terms of the initial and target data is positive semi-definite. Background The Nevanlinna–Pick theorem represents an n-point generalization of the Schwarz lemma. The invariant form of the Schwarz lemma states that for a holomorphic function f:\mathbb\to\mathbb, for all \lambda_1, \lambda_2 \in \mathbb, : \left, \frac\ \leq \left, \frac\. Setting ... [...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] |
Levinson Recursion
Levinson recursion or Levinson–Durbin recursion is a procedure in linear algebra to recursively calculate the solution to an equation involving a Toeplitz matrix. The algorithm runs in time, which is a strong improvement over Gauss–Jordan elimination, which runs in Θ(''n''3). The Levinson–Durbin algorithm was proposed first by Norman Levinson in 1947, improved by James Durbin in 1960, and subsequently improved to and then multiplications by W. F. Trench and S. Zohar, respectively. Other methods to process data include Schur decomposition and Cholesky decomposition. In comparison to these, Levinson recursion (particularly split Levinson recursion) tends to be faster computationally, but more sensitive to computational inaccuracies like round-off errors. The Bareiss algorithm for Toeplitz matrices (not to be confused with the general Bareiss algorithm) runs about as fast as Levinson recursion, but it uses space, whereas Levinson recursion uses only ''O''(''n'') space. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carathéodory Function
In mathematical analysis, a Carathéodory function (or Carathéodory integrand) is a multivariable function that allows us to solve the following problem effectively: A composition of two Lebesgue-measurable functions does not have to be Lebesgue-measurable as well. Nevertheless, a composition of a measurable function with a continuous function is indeed Lebesgue-measurable, but in many situations, continuity is a too restrictive assumption. Carathéodory functions are more general than continuous functions, but still allow a composition with Lebesgue-measurable function to be measurable. Carathéodory functions play a significant role in calculus of variation, and it is named after the Greek mathematician Constantin Carathéodory. Definition W:\Omega\times\mathbb^\rightarrow\mathbb\cup\left\ , for \Omega\subseteq\mathbb^ endowed with the Lebesgue measure, is a Carathéodory function if: 1. The mapping x\mapsto W\left(x,\xi\right) is Lesbegue-measurable for every \xi\in\m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad'' − ''bc'' ≠ 0. Geometrically, a Möbius transformation can be obtained by first performing stereographic projection from the plane to the unit two-sphere, rotating and moving the sphere to a new location and orientation in space, and then performing stereographic projection (from the new position of the sphere) to the plane. These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle. The Möbius transformations are the projective transformations of the complex projective line. They form a group called the Möbius group, which is the projective linear group PGL(2,C). Together with its subgroups, it has numerous applications in mathematics and physics. Möbius transfor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continued Fraction
In mathematics, a continued fraction is an expression (mathematics), expression obtained through an iterative process of representing a number as the sum of its integer part and the multiplicative inverse, 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 (mathematics), infinite expression. In either case, all integers in the sequence, other than the first, must be positive number, 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 function (mathematics), functions are used in place of one or more of the numerat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Polynomials On The Unit Circle
In mathematics, orthogonal polynomials on the unit circle are families of polynomials that are orthogonal with respect to integration over the unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ... in the complex plane, for some probability measure on the unit circle. They were introduced by . Definition Suppose that \mu is a probability measure on the unit circle in the complex plane, whose support is not finite. The orthogonal polynomials associated to \mu are the polynomials \Phi_n(z) with leading term z^n that are orthogonal with respect to the measure \mu. The Szegő recurrence Szegő's recurrence states that :\Phi_0(z) = 1 :\Phi_(z)=z\Phi_n(z)-\overline\alpha_n\Phi_n^*(z) where :\Phi_n^*(z)=z^n\overline is the polynomial with its coefficients reversed and comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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(z) + \rho_\phi^*(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 3-term recurrence relation is a sequence of orthogonal polynomials. The theorem was introduced in the theory of ... References * * G. Szegő, "Orthogonal polynomials", Colloq. Publ., 33, Amer. Math. Soc. (1967) Orthogonal polynomials {{math-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
