Gaussian Analytic Functions in the Polydisk

We study hyperbolic Gaussian analytic functions in the unit polydisk of \(\mathbb {C}^n\). Following the scheme previously used in the unit ball, we first study the asymptotics of fluctuations of linear statistics as the directional intensities \(L_j\), \(j=1,\dots ,n\) tend to \(\infty \). Then, we estimate the probability of large deviations of such linear statistics and use the estimate to prove a hole theorem. Our proofs are inspired by the methods of M. Sodin and B. Tsirelson for the one-dimensional case, and B. Shiffman and S. Zelditch for the study of the analogous problem for compact Kähler manifolds.     

Gaussian Versus Optimal Integration of Analytic Functions

We consider error estimates for optimal and Gaussian quadrature formulas if the integrand is analytic and bounded in a certain complex region. First, a simple technique for the derivation of lower bounds for the optimal error constants is presented. This method is applied to Szeg?-type weight functions and ellipses as regions of analyticity. In this situation, the error constants for the Gaussian formulas are close to the obtained lower bounds, which proves the quality of the Gaussian formulas and also of the lower bounds. In the sequel, different regions of analyticity are investigated. It turns out that almost exclusively for ellipses, the Gaussian formulas are near-optimal. For classes of simply connected regions of analyticity, which are additionally symmetric to the real axis, the asymptotic of the worst ratio between the error constants of the Gaussian formulas and the optimal error constants is calculated. As a by-product, we prove explicit lower bounds for the Christoffel-function for the constant weight function and arguments outside the interval of integration. September 7, 1995. Date revised: October 25, 1996.     

The Remainder Term for Analytic Functions of Symmetric Gaussian Quadratures

For analytic functions the remainder term of Gaussian quadrature rules can be expressed as a contour integral with kernel . In this paper the kernel is studied on elliptic contours for a great variety of symmetric weight functions including especially Gegenbauer weight functions. First a new series representation of the kernel is developed and analyzed. Then the location of the maximum modulus of the kernel on suitable ellipses is determined. Depending on the weight function the maximum modulus is attained at the intersection point of the ellipse with either the real or imaginary axis. Finally, a detailed discussion for some special weight functions is given.

     

线性同胚于星象函数的一族解析函数

本文定义了线性同胚于星象函数的－族解析函数Ａ（，α）．我们导出Ａ（α）中函数的积分表达式：借助算子理论研究Ａ（，α）族的包含关系并确定它的闭凸包、闭凸包的极值点和它的支撑点；利用一个阶微分从属证明关于实部的二个不等式．最后，我们还证明Ａ（，α）中函数的偏差定理．     

一类新的解析函数族

定义了一族解析函数B(λ,α,β),导出该族中函数的积分表达式;借助算子理论建立B(λ,α,β)的包含关系,讨论端点性质;由此推出族中函数的偏差定理.     

Universality Property and Dimension for Frames

Order - The universality property plays an important role in the field of frames and the notion of saturated class of frames is combined with this property (see Dube et al. (Topology and its...     

A Uniqueness Theorem for Bounded Analytic Functions

Suppose that K is a linear space of functions analytic in somedomain D in the complex plane. A sequence = (_k) of distinctpoints from D is said to be a set of uniqueness for K if fKand f(_k) = 0 for all k imply f0. Depending on the dispersionand the density of on the one hand, and the growth of the functionsin K on the other, one may often require only |f(_k)| a_k forsome sequence of positive numbers a_k, and still conclude thatf0 for fK. Of particular interest are sharp conditions on thedecay of a_k, which reflect the interplay between growth anddecay of analytic functions. 1991 Mathematics Subject Classification30A99, 31A05.     

Analytic Extension of Functions from Analytic Hilbert Spaces

Let M be an invariant subspace of Hv2. It is shown that for each f∈M⊥, f can be analytically extended across (?)Bd\σ(Sz1,…, Szd).     

A Property of Piecewise Smooth Functions

Piecewise smooth equations are increasingly important in the numerical treatment of complementarity problems and models of equilibrium. This note brings out a property of the functions that enter such equations, for instance through penalty expressions.     

Covering Dimension and Universality Property on Frames

Order - Studies such as Dube et al. (Topology Appl. 160(18):2454–2464 2013), Georgiou et al. (Algebra Universalis 80(2):16, 2019), Iliadis (2005; Topology Appl. 179, 99–110, 2015;...     

Linear Generalized Analytic Functions

Complex-valued continuous functions are approximated using solutions of complex partial differential equations of Vekua type.     

On Expanding Neighborhoods of Local Universality of Gaussian Unitary Ensembles

The classical universality theorem states that the Christoffel–Darboux kernel of the Hermite polynomials scaled by a factor of \(1/\sqrt n\) tends to the sine kernel in local variables \(\tilde x,\tilde y\) in a neighborhood of a point \(x^*\in(-\sqrt 2,\sqrt 2)\)). This classical result is well known for \(\tilde x,\tilde y\in{K}\Subset\mathbb{R}\). In this paper, we show that this classical result remains valid for expanding compact sets K = K(n). An interesting phenomenon of admissible dependence of the expansion rate of compact sets K(n) on x* is established. For \(x^*\in(-\sqrt 2,\sqrt 2)\backslash\left\{0\right\}\)) and for x* = 0, there are different growth regimes of compact sets K(n). A transient regime is found.     

Stable Extrapolation of Analytic Functions

Foundations of Computational Mathematics - This paper examines the problem of extrapolation of an analytic function for $$x > 1$$ given $$N+1$$ perturbed samples from an equally spaced grid...     

Analytic Extension of Smooth Functions

Let F be a closed proper subset of ?ⁿ and let ?* be a class of ultradifferentiable functions. We give a new proof for the following result of Schmets and Valdivia on analytic modification of smooth functions: for every function ? ∈ ?* (?ⁿ) there is ${\widetilde f} \in {\cal E}_{*}(\rm R ^{n})$ which is real analytic on ?ⁿF and such that ?^a ? ¦ _F = ?^a ? ¦ _F for any a ∈ ?₀ ⁿ. For bounded ultradifferentiable functions ? we can obtain ${\widetilde f}$ by means of a continuous linear operator.     

线性同胚于星象函数的一族解析函数(II)

本文继续研究[1]中引进的线性同胚于星象函数的一族解析函数A(λ,α),给出了A(λ,α)族的卷积定理以及系数和的一个不等式.     

线性同胚于星象函数的一族解析函数（Ⅱ）

本文继续研究了「１」中引进的线性同胚于星象函数的一族解析函数Ａ（λ,α）,给出了Ａ（λ,α）族的卷积定理以及系数和的一个不等式。