Optimal Recovery of the Derivative of Periodic Analytic Functions from Hardy Classes

LetS_β{z : |Im z|<β}. For 2π-periodic functions which are analytic inS_βwithp-integrable boundary values, we construct an optimal method of recovery off′(ξ), ξS_β, using information about the valuesf(x₁), mldr;, f(x_n), x_j[0, 2π).     

Hankel Operators on Bounded Analytic Functions

For a domain in the complex plane and a bounded measurable function on , the generalized Hankel operator on is the operator of multiplication by followed by projection into . Under certain conditions on we show that either is compact or there is an embedded on which is bicontinuous. We characterize those 's for which is compact in the case that is a Behrens roadrunner domain.

     

Joint Spectra of Toeplitz Operators and Optimal Recovery of Analytic Functions

We consider optimal recovery problems in weighted Bergman spaces in the unit ball of ? ^k . It is shown that if corresponding measures are Carleson and not compactly supported, the optimal recovery algorithm is a limit of optimal recovery algorithms corresponding to truncated problems with compactly supported measures. We also express the Lagrange equation for the dual problem in terms of joint spectra of Toeplitz operators induced by information measures and use this expression for proving that the regularity condition in the dual problem is stable. Examples in the last section illustrate our results.     

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.     

Sufficient Conditions for Bounded Turning of Analytic Functions

Ukrainian Mathematical Journal - Consider a function f analytic in the open unit disk and normalized so that f(0) = f′(0) ? 1 = 0. The methods of the theory of first-order differential...     

Uniform Approximation by the Class of Functions with Bounded Derivative

Mathematical Notes - In the paper, the problem of uniform approximation of a continuous function defined on an interval is considered. The approximating functions have absolutely continuous...     

Hadamard and Schwarz Type Theorems and Optimal Recovery in Spaces of Analytic Functions

We prove a general theorem which gives a necessary condition of extremum in the dual optimal recovery problem in terms of inclusion in certain annihilators. Applications of this theorem yield Hadamard and Schwarz type results. We also construct related optimal recovery methods.     

A Uniqueness Theorem for Bounded Analytic Functions on the Polydisc

Given n, N ≥ 1 we construct a set of points ${\lambda_1,{\ldots},\lambda_{N^n}\in{\mathbb D}^n}$ such that for each rational inner function f on ${{\mathbb D}^n}$ of degree less than N the Pick problem on ${{\mathbb D}^n}$ with data ${\lambda_1,{\ldots},\lambda_{N^n}}$ and ${f(\lambda_1),{\ldots},f(\lambda_{N^n})}$ has a unique solution. In particular, we construct a 1-dimensional inner variety V and show that the points ${\lambda_1,{\ldots},\lambda_{N^n}}$ may be chosen almost arbitrarily in ${V\cap{\mathbb D}^n}$ . Our results state that f is uniquely determined in the Schur class of ${{\mathbb D}^n}$ by its values on ${\lambda_1,{\ldots},\lambda_{N^n}}$ .     

Univalent Criteria for Analytic Functions Involving Schwarzian Derivative

In this paper, some new criteria for univalence of analytic functions de-fined in the unit disk in terms of two parameters are presented. Moreover, the related result of Aharonov and Elias(Aharonov D, Elias U. Univalence criteria depending on parameters. Anal. Math. Phys., 2014, 4(1-2): 23–34) is generalized.     

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.     

Stable Recovery of Analytic Functions using Basic Hypergeometric Series

Two stable sampling formulas for reconstructing analytic functions from exponentially spaced samples are considered. Criterion for selecting regularization parameter and error estimates are obtained.     

R1上有界函数的导数零点存在性问题

研究定义域为R 1 上有界函数的n阶导数零点存在性问题及一般结论.     

模糊有界变差函数全变差的积分表示与距离导数

定义和讨论了模糊数值函数的距离导数,给出了模糊有界变差函数全变差的积分表示.发现模糊绝对连续函数是几乎处处距离可导的,距离导数的积分等于其原函数的总变差,从而给出了模糊有界变差函数全变差的积分表示.     

Koch曲线所围区域内的一类解析函数边值问题

在经典解析函数边值理论中,当L为复平面上逐段光滑封闭曲线时,在L所围的内部和外部,Cauchy型积分解析;通过对Cauchy主值积分的讨论,可得Cauchy型积分在L上的左、右边值,且边值满足Plemelj公式.基于Koch曲线的构造方法,对一系列Cauchy型积分取极限,并附加上一定的Hlder条件,可得在Koch曲线所围的内部和外部区域内都解析的Cauchy型积分函数,进一步得到与经典解析函数边值问题类似的结果.     

Spectral Representation of Analytic Diagonalizable Matrix-valued Functions that Commute with their Derivative

In 1981, Goff showed that an analytic n × n self-adjoint matrix-valued function A defined on an open interval I of the real axis that commutes with its derivative must be functionally commutative. We generalize Goff's result by showing that it remains valid for diagonalizable matrices and that we need only require this diagonalizability property on an open subinterval I o of I . We also show that diagonalizability property propagates from the subinterval I o to the whole interval I and we obtain the spectral representation of the function A and its derivatives D j A , j = 0,1,2, < .     

Bounded Cosine Functions Close to Continuous Scalar Bounded Cosine Functions

Let \({(C(t))_{t\in \mathbb R}}\) be a cosine function in a unital Banach algebra. We show that if \({\sup_{t\in \mathbb R}\Vert C(t)-c(t)\Vert < 2}\) for some continuous scalar bounded cosine function \({(c(t))_{t\in \mathbb R},}\) then the closed subalgebra generated by \({(C(t))_{t\in \mathbb R}}\) is isomorphic to \({\mathbb C^k}\) for some positive integer k. If, further, \({\sup_{t\in \mathbb R}\Vert C(t)-c(t)\Vert < \frac{8}{3\sqrt 3},}\) then \({C(t)=c(t)}\) for \({t\in \mathbb R}\).     

Bounded Cyclic Functions in the Ball

With the help of the H²-corona theorem we obtain a sufficient condition for cyclicity in the Bergman space A¹ in the ball. Bibliography: 5 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 102–110.     

多元 Besov-Wiener 类的无穷维宽度和最优恢复

该文考虑 Besov-Wiener 类S_pqθ^r B(R^d)和 S_pqθ^r B(R^d)在 L_q(R^d) 空间下 (1≤q≤ p <∞ ) 的无穷维σ -宽度和最优恢复问题.通过考虑样条函数逼近和构造一种连续样条算子, 得到了关于无穷维Kolmogorov 宽度、无穷维线性宽度、无穷维 Gel'fand 宽度和最优恢复的弱渐近结果.