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Trigonometric Moment Problem
In mathematics, the trigonometric moment problem is formulated as follows: given a sequence \_, does there exist a distribution function \mu on the interval ,2\pi/math> such that: c_k = \frac\int_0 ^ e^\,d \mu(\theta), with c_ = \overline_k for k \geq 1. In case the sequence is finite, i.e., \_^, it is referred to as the truncated trigonometric moment problem. An affirmative answer to the problem means that \_ are the Fourier-Stieltjes coefficients for some (consequently positive) Radon measure \mu on ,2\pi/math>. Characterization The trigonometric moment problem is solvable, that is, \_^ is a sequence of Fourier coefficients, if and only if the Hermitian Toeplitz matrix T = \left(\begin c_0 & c_1 & \cdots & c_n \\ c_ & c_0 & \cdots & c_ \\ \vdots & \vdots & \ddots & \vdots \\ c_ & c_ & \cdots & c_0 \\ \end\right) with c_=\overline for k \geq 1, is positive semi-definite. The "only if" part of the claim ... [...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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle Group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \. The circle group forms a subgroup of \mathbb C^\times, the multiplicative group of all nonzero complex numbers. Since \mathbb C^\times is abelian, it follows that \mathbb T is as well. A unit complex number in the circle group represents a rotation of the complex plane about the origin and can be parametrized by the angle measure \theta: \theta \mapsto z = e^ = \cos\theta + i\sin\theta. This is the exponential map for the circle group. The circle group plays a central role in Pontryagin duality and in the theory of Lie groups. The notation \mathbb T for the circle group stems from the fact that, with the standard topology (see below), the circle group is a 1-torus. More generally, \mathbb T^n (the direct product of \mathbb T with i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Problems
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty."Kendall's Advanced Theory of Statistics, Volume 1: Distribution Theory", Alan Stuart and Keith Ord, 6th Ed, (2009), .William Feller, ''An Introduction to Probability Theory and Its Applications'', (Vol 1), 3rd Ed, (1968), Wiley, . The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ("heads" and "tails") are both equally probable; the probability of "heads" equals the probability of "tails"; and since no other outcomes are possible, the probability of either "heads" or "tails" is 1/2 (which could also be written as 0.5 or 50%). These conc ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiener's Lemma
In mathematics, Wiener's lemma is a well-known identity which relates the asymptotic behaviour of the Fourier coefficients of a Borel measure on the circle to its atomic part. This result admits an analogous statement for measures on the real line. It was first discovered by Norbert Wiener. Statement * Given a real or complex Borel measure \mu on the unit circle \mathbb T, let \mu_a=\sum_j c_j\delta_ be its atomic part (meaning that \mu(\)=c_j\neq 0 and \mu(\)=0 for z\not\in\. Then :\lim_\frac\sum_^N, \widehat\mu(n), ^2=\sum_j, c_j, ^2, where \widehat(n)=\int_z^\,d\mu(z) is the n-th Fourier coefficient of \mu. * Similarly, given a real or complex Borel measure \mu on the real line \mathbb R and called \mu_a=\sum_j c_j\delta_ its atomic part, we have :\lim_\frac\int_^R, \widehat\mu(\xi), ^2\,d\xi=\sum_j, c_j, ^2, where \widehat(\xi)=\int_e^\,d\mu(x) is the Fourier transform of \mu. Proof * First of all, we observe that if \nu is a complex measure on the circle then :\frac\sum_^N\w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Szegő Limit Theorems
In mathematical analysis, the Szegő limit theorems describe the asymptotic behaviour of the determinants of large Toeplitz matrices. They were first proved by Gábor Szegő. Notation Let \phi: \mathbb\to\mathbb be a complex function ("''symbol''") on the unit circle. Consider the n\times n Toeplitz matrices T_n(\phi), defined by : T_n(\phi)_ = \widehat\phi(k-l), \quad 0 \leq k,l \leq n-1, where : \widehat\phi(k) = \frac \int_0^ \phi(e^) e^ \, d\theta are the Fourier coefficients of \phi. First Szegő theorem The first Szegő theorem states that, if \phi>0 and \phi\in L_1(\mathbb), then The right-hand side of () is the geometric mean of \phi (well-defined by the arithmetic-geometric mean inequality). Second Szegő theorem Denote the right-hand side of () by G. The second (or strong) Szegő theorem asserts that if, in addition, the derivative of \phi is Hölder continuous Hölder: * ''Hölder, Hoelder'' as surname * Hölder condition * Hölder's inequality * Hölder ... [...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 In mathematical analysis, a Carathéodory function (or Carathéodory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Measure
In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theorem. It has had important applications in quantum mechani ... [...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 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 complex conjugated, and where the Verblunsky coefficients \alpha_n are complex numbers with absolute values less than 1. Verblunsky's theorem Verblunsky's theorem states that any sequence of complex numbers in the open unit disk is the sequence of Verblunsky ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moment Problem
In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure ''μ'' to the sequences of moments :m_n = \int_^\infty x^n \,d\mu(x)\,. More generally, one may consider :m_n = \int_^\infty M_n(x) \,d\mu(x)\,. for an arbitrary sequence of functions ''M''''n''. Introduction In the classical setting, μ is a measure on the real line, and ''M'' is the sequence . In this form the question appears in probability theory, asking whether there is a probability measure having specified mean, variance and so on, and whether it is unique. There are three named classical moment problems: the Hamburger moment problem in which the support of μ is allowed to be the whole real line; the Stieltjes moment problem, for , +∞); and the Hausdorff moment problem for a bounded interval, which without loss of generality may be taken as , 1 Existence A sequence of numbers ''m''''n'' is the sequence of moments of a measure ''μ'' if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamburger Moment Problem
In mathematics, the Hamburger moment problem, named after Hans Ludwig Hamburger, is formulated as follows: given a sequence (''m''0, ''m''1, ''m''2, ...), does there exist a positive Borel measure ''μ'' (for instance, the measure determined by the cumulative distribution function of a random variable) on the real line such that :m_n = \int_^\infty x^n\,d \mu(x) \text In other words, an affirmative answer to the problem means that (''m''0, ''m''1, ''m''2, ...) is the sequence of moments of some positive Borel measure ''μ''. The Stieltjes moment problem, Vorobyev moment problem, and the Hausdorff moment problem are similar but replace the real line by ,+\infty) (Stieltjes and Vorobyev; but Vorobyev formulates the problem in the terms of matrix theory), or a bounded interval (Hausdorff). Characterization The Hamburger moment problem is solvable (that is, (''m''''n'') is a sequence of moments) if and only if the corresponding Hankel kernel on the nonnegative integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bochner's Theorem
In mathematics, Bochner's theorem (named for Salomon Bochner) characterizes the Fourier transform of a positive finite Borel measure on the real line. More generally in harmonic analysis, Bochner's theorem asserts that under Fourier transform a continuous positive-definite function on a locally compact abelian group corresponds to a finite positive measure on the Pontryagin dual group. The case of sequences was first established by Gustav Herglotz (see also the related Herglotz representation theorem.) The theorem for locally compact abelian groups Bochner's theorem for a locally compact abelian group ''G'', with dual group \widehat, says the following: Theorem For any normalized continuous positive-definite function ''f'' on ''G'' (normalization here means that ''f'' is 1 at the unit of ''G''), there exists a unique probability measure ''μ'' on \widehat such that : f(g) = \int_ \xi(g) \,d\mu(\xi), i.e. ''f'' is the Fourier transform of a unique probability measure ''μ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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