Finite Positive Borel Measure Research Articles

On a generalization of the trigonometric moment problem

The trigonometric moment problem asks if there exists a positive finite Borel measure on the unit circle whose first n trigonometric moments take some specified values. This paper treats the case when only some of the first n trigonometric moments are specified. A solution of the problem is found when two of the moments are specified and they are real. If the moments are complex numbers, some partial results are obtained. In the case when all the n first moments except the n − 1th are specified, a complete solution of the problem is presented.

Strong asymptotics on the support of the measure of orthogonality for polynomials orthogonal with respect to a discrete Sobolev inner product

In (6, 8) relative asymptotics for Sobolev orthogonal polynomials were studied. When the continuous part of the measure satisfies the Szegýo condition, the strong asymptotics outside the support of the measure follow from (6, 8). In this paper, we analyze this problem on the support of the measure. 1. Introduction and statements of results Some problems arising from approximation theory, mathematical physics, and number theory have motivated different generalizations of the standard notion of orthogonality. If µ is a finite positive Borel measure supported on the real line, then {pn} is the sequence of standard orthonormal polynomials (OP) with respect to µ when

Riesz's Functions in Weighted Hardy and Bergman Spaces

AbstractLet μ be a finite positive Borel measure on the closed unit disc . For each a in , put where ƒ ranges over all analytic polynomials with f(a) = 1. This upper semicontinuous function S(a) is called a Riesz's function and studied in detail. Moreover several applications are given to weighted Bergman and Hardy spaces.

On A Minimization Problem for Vector Fields in L 1

This paper treats the problem of minimizing the norm of vector fields in L1 with prescribed divergence. The ridge of Ω. plays an important role in the analysis, and in the case where Ω ⊂ R2 is a polygonal domain, the ridge is thoroughly analysed and some examples are presented. In the case where Ω ⊂ Rn is a Lipschitz domain and the divergence is a finite positive Borel measure, the infimum is calculated, and it is shown that if an extremal exists, then it is of the form υ1 = −F▿d, where F is a nonnegative function and d(x) is the distance from x ∈ Ω to the boundary ∂Ω. Finally, if Ω ⊂ R2 is a polygonal domain and the measure is represented by a nonnegative continuous function, then an explicit expression for the extremal is given, and it is proven that this extremal is unique.

Mixing coefficient, generalized maximal correlation coefficients, and weakly positive measures

Let μ be a positive finite Borel measure on the real line R. For t ≥ 0 let e t · E 1 and E 2 denote, respectively, the linear spans in L 2( R, μ) of { e isx , s > t} and { e isx , s < 0}. Let θ: R → C such that |theta;| = 1, denote by α t ( θ, μ) the angle between θ · e t · E 1 and E 2. The problems considered here are that of describing those measures μ for which (1) α t ( θ, μ) > 0, (2) α t(θ, μ) → π 2 as t → ∞ (such μ arise as the spectral measures of strongly mixing stationary Gaussian processes), and (3) give necessary and sufficient conditions for the rate of convergence of the generalized maximal correlation coefficient: ϱ t ( θ, μ) = cos α t ( θ, μ). Using this coefficient we characterize the stationary continuous processes that are (a) completely regular and (b) strongly mixing Gaussian. We also give necessary and sufficient conditions for the rate of convergence of (a) the maximal correlation coefficient and (b) the mixing coefficient in the Gaussian case.

Rates of convergence in a central limit theorem for stochastic processes defined by differential equations with a small parameter

Let μ be a positive finite Borel measure on the real line R. For t ≥ 0 let e t · E 1 and E 2 denote, respectively, the linear spans in L 2( R, μ) of { e isx , s > t} and { e isx , s < 0}. Let θ: R → C such that ∥θ∥ = 1, denote by α t ( θ, μ) the angle between θ · e t · E 1 and E 2. The problems considered here are that of describing those measures μ for which (1) α t ( θ, μ) > 0, (2) α t(θ, μ) → π 2 as t → ∞ (such μ arise as the spectral measures of strongly mixing stationary Gaussian processes), and (3) give necessary and sufficient conditions for the rate of convergence of the generalized maximal correlation coefficient: ϱ t ( θ, μ) = cos α t ( θ, μ). Using this coefficient we characterize the stationary continuous processes that are (a) completely regular and (b) strongly mixing Gaussian. We also give necessary and sufficient conditions for the rate of convergence of (a) the maximal correlation coefficient and (b) the mixing coefficient in the Gaussian case.

A FORMULA FOR THE OPTIMAL VALUE IN THE MONGE-KANTOROVICH PROBLEM WITH A SMOOTH COST FUNCTION, AND A CHARACTERIZATION OF CYCLICALLY MONOTONE MAPPINGS

The general Monge-Kantorovich problem consists in the computation of the optimal value where the cost function and the measure on with are assumed to be given, is the cone of finite positive Borel measures on , and and are the projections on the first and second coordinates, which assign to a measure the corresponding marginal measures. An explicit formula is obtained for in the case when is a domain in and is bounded, vanishes on the diagonal, and is continuously differentiable in a neighborhood of the diagonal. Conditions for the set to be nonempty are investigated, and with their help new characterizations of cyclically monotone mappings are obtained.

Asymptotics for Lp extremal polynomials on the unit circle

Let p > 1, and dμ a positive finite Borel measure on the unit circle Γ: = {z ϵ C: ¦z¦ = 1} . Define the monic polynomial φ n, p ( z)= z n +… ϵP n >(the set of polynomials of degree at most n) satisfying ∫ Г |φ n, p(z)| p PϵPn-1 inf ∫ Г |z n + P| p dμ . Under certain conditions on dμ, the asymptotics of φ n, p ( z) for z outside, on, or inside Γ are obtained (cf. Theorems 2.2 and 2.4). Zero distributions of φ n, p are also discussed (cf. Theorems 3.1 and 3.2).

Szego's Extremum Problem on the Unit Circle

It is shown that the Christoffel functions arising from the Szegd extremum problem associated with a finite positive Borel measure on the interval [- wr, wr) satisfy

Perturbations in Toeplitz matrices. II. Asymptotic properties

Given dμ a finite positive Borel measure on the interval [0, 2π]with an infinite set as its support, ( φ n ( z)) n = 0 ∞ the monic orthogonal polynomials, and ( ϑ n ( z)) n = 0 ∞ the orthonormal polynomials on the unit circle associated with this measure, we determine conditions in the measure and in the parameter t in order that the sequences ( φ n ( t)) n = 0 ∞ and ( ϑ n ( t)) n = 0 ∞ do not belong to l 2. This allows us to study the asymptotic behaviour of the ratio ( k n α n ) for the leading coefficients of ϑ n ( z) and P n ( z) with ( P n ( z)) n = 0 ∞ the system of orthonormal polynomials with respect to the inner product 〈P(z),Q(z)〉= 1 2π ∫ 0 2π P(z) Q(z) dμ(0)+P(t) Q(t), z=e i0, with tϵC

A representation theorem for cyclic analytic two-isometries

A bounded linear operator T on a complex separable Hilbert space Z is called a 2-isometry if T*2T2-2T* T+I = O . We say that T is analytic if nn,0 Tnt = (0) . In this paper we show that every cyclic analytic 2-isometry can be represented as multiplication by z on a Dirichlet-type space D(8). Here ,ze denotes a finite positive Borel measure on the unit circle. For two measures ,u and v the 2-isometries obtained as multiplication by z on D(8) and D(v) are unitarily equivalent if and only if ,u = v . We also investigate similarity and quasisimilarity of these 2-isometries, and we apply our results to the invariant subspaces of the Dirichlet shift.

On Nevai's characterization of measures with almost everywhere positive derivative

It is known that if dμ is a finite positive Borel measure on the unit circle ∂Δ := {z ϵ C:¦z¦ = 1} with μ′ > 0 a.e. in [0, 2π], then lim n→∞ ∫ 2π 0 |ϑ n(z)| 2 | n+l(z)| 2 −1 dθ=0, z=e iθ (∗) uniformly in l ⩾ 0, where { ϑ n ( z)} n = 0 ∞ is the set of orthonormal polynomials with respect to dμ. Recently, Nevai proved that if (∗) holds uniformly in l ⩾ 0, then μ′ > 0 a.e. in [0, 2π]. In this note we establish another characterization of such measures in terms of Turán-type inequalities and we utilize it to give an alternative proof of Nevai's result.

Asymptotics of polynomials orthogonal with respect to varying measures

Letdσ be a finite positive Borel measure on the interval [0, 2π] such that σ′>0 almost everywhere; andW n be a sequence of polynomials, degW n =n, whose zeros (w n ,1,⋯,w n,n lie in [|z|≤1]. Let ∥dσ n ∥<+∞ for eachn∈N, wheredσ n =dσ/|W n (e iθ )|2. We consider the table of polynomialsϕ n,m such that for each fixedn∈N the systemϕ n,m,m∈N, is orthonormal with respect todσ n . If $$\mathop {\lim }\limits_n \sum\limits_{\iota = 1}^n {(1 - |w_{n,\iota } |) = + \infty }$$ andk∈N then lim n ϕ n,n+k+1(w)/ϕ n,n+k (w)=w uniformly on each compact set contained in [|w|≥1]. This result extends a well-known one of E. A. Rakhmanov. Extensions of several results of A. Mate, P. Nevai, and V. Totik are also obtained; e.g., the above conditions also yield $$\mathop {\lim }\limits_n \int {(\sqrt {\sigma '(\theta )} } |\varphi _{n,n + k} (e^{i\theta } )/W_n (e^{i\theta } )| - 1)^2 d\theta = 0$$ which enables us to restate much of Szego's theory in this new setting. Weak convergence results of orthogonal polynomials on the real line are also obtained.

Tangential H∞-images of boundary curves

1.Introduction. Suppose that Ω is a region (i.e. a connected open set) in ࠶n, for some fixedn≥ 1. We define (Γ, μ) to be aFatou pair inΩ if(a) Γ is a continuous family of boundary curves γwin Ω, one ending at each w ∈ ∂Ω,(b) μ is a positive finite Borel measure on ∂Ω, and(c) the conclusion of Fatou's theorem holds with respect to Γ and μ. Let us state (a) and (c) in more detail:(a) The map(w, t) → γw(t)is continuous, from ∂Ω × [0, 1) into Ω, andfor every w in the boundary ∂Ω of Ω.(c) For everyf ∈ H∞(Ω)(the class of all bounded holomorphic functions in Ω), the limitexists a.e. [μ].

Amenable groups and stabilizers of measures on the boundary of a Hadamard manifold

Let X be a symmetric space of noncompact type or more generally a Hadamard manifold, i.e. a complete simply connected Riemannian manifold of nonpositive sectional curvature. We consider a finite positive Borel-measure # on the ideal boundary X(~) (see Sect. 2 for the definitions). Let G be the isometry group of X and G~ the subgroup which stabilizes the measure #. In the case of a symmetric space we obtain the following result.

Multipliers of Bergman spaces into Lebesgue spaces

Let U be the open unit disk in the complex plane endowed with normalized Lebesgue measure m. will denote the usual Lebesgue space with respect to m, with 0&lt;p&lt;+∞. The Bergman space consisting of the analytic functions in will be denoted . Let μ be some positivefinite Borel measure on U. It has been known for some time (see [6] and [9]) what conditions on μ are equivalent to the estimate: There is a constant C such thatprovided 0&lt;p≦q.

Weak convergence of measures and weak type (1,𝑞) of maximal convolution operators

Let G ∗ {G^ * } be the maximal convolution operator associated with a sequence of L 1 {L^1} kernels. We show that if G ∗ {G^ * } is of weak type ( 1 , q ) (1,q) , 1 ≤ q &gt; ∞ 1 \leq q &gt; \infty , over a subset N {\mathcal N} of M {\mathcal M} (the space of all finite positive Borel measures on R h {{\bf {R}}^h} endowed with the weak topology), then G ∗ {G^ * } is of weak type ( 1 , q ) (1,q) over the closed cone in M {\mathcal M} generated by N {\mathcal N} . As a particular case we obtain a well-known result by de Guzman.

Hörmander's Carleson theorem for the ball

Let denote the unit ball in ℂ2 and let Sdenote its boundary, the unit sphere. For z ∈ B and δ&gt;0, the following non isotropic balls are defined, where A finite positive Borel measure μ, on B is called a Carleson measure if there exists a constant C for whichHere σ denotes normalized surface area measure on S. The following theorem was obtained by Hörmander [6] as a special case of more general variants for strictly pseudoconvex domains in ℂn. Recently Cima and Wogen [3] derived it from a Carleson measure theorem for Bergman spaces of the ball. A different direct approach to the Bergman context, and related settings, is given in Leucking [7].

On Some Integral Equations with Locally Finite Measures and $L^\infty $-Perturbations

Let $g \in C(R),f \in L_{{\text{loc}}}^\infty (R^ + )$ and let $\mu $ be a real locally finite positive definite Borel measure on $R^ + $. We investigate a relation between the solution of the nonlinear scalar Volterra equation \[ x'(t) + \int_{[0,t]} {g(x(t - s))} d\mu (s) = f(t),\quad t \in R^ + ,\quad x(0) = x_0 ,\] and the solution of the linear equation with the same data \[ z'(t) + \int_{[0,t]} {z(t - s)} d\mu (s) = f(t),\quad t \in R^ + ,\quad z(0) = x_0 .\] This relation, when combined with results (established in this paper) on the set of bounded solutions of certain limit equations \[ y(t) + \int_{R^ + } {g(y(t - s))} a(s)ds = 0,\quad t \in R,\] allows us to obtain new asymptotic results for $x(t)$ in the case when both $\mu $ and f are large in a precise sense.

The Poisson Integral of a Singular Measure

Let σ be a finite positive singular Borel measure defined on Euclidean space RN. For w ∈ RN and y &gt; 0, its Poisson integral is defined by the formulawhere CN is chosen so thatSince σ is singular, almost everywhere with respect to Lebesgue measure on RN. On the other hand, almost everywhere dσ. It follows that for all sufficiently small y,is a non-empty open subset of RN. If σ has compact support then |Ey| → 0 as y → 0, where |Ey| denotes the Lebesgue measure of Ey. In this paper we give a lower bound on the rate at which |Ey| may go to zero. The lower bound depends on the smoothness of the measure; the smoother the measure, the more slowly |Ey| may approach 0.