Iteration of functions analytic on a disk

Aequationes mathematicae -     

Wiener-Hopf operators and generalized analytic functions

An almost periodic generalization of H+C is defined and analyzed. Applications of this analysis are made to the type II index theory of Wiener-Hopf operators with almost periodic symbols.Supported in part by the National Science Foundation.     

Integral operators preserving certain analytic functions

The purpose of the present paper is to investigate some integral preserving properties for certain analytic functions in the open unit disk. We also obtain the conditions for close-to-convex functions in a sector. Our results contain some interesting corollaries as the special cases.     

Abel wave operators. II. Wave operators for functions of operators

It is shown that in a scattering theory with averaging the invariance principle of M. S. Birman and T. Kato (in its strong form) is always valid.     

Toeplitz operators and arguments of analytic functions

For a Toeplitz operator T _φ , we study the interrelationship between smoothness properties of the symbol φ and those of the functions annihilated by T _φ . For instance, it follows from our results that if φ is a unimodular function on the circle lying in some Lipschitz or Zygmund space Λ^α with 0 < α < ∞, and if f is an H ^p -function (p ≥ 1) with T _φ f = 0, then f ∈ Λ^α and

Factorization of fredholm operators on analytic functions

Let Ω be a simply connected domain in the complex plane, and $\text{A(Ω}{}^{\mathrm{n}}\text{)}$, the space of functions which are defined and analytic on $\text{Ω}{}^{\mathrm{n}}$, if K is the operator on elements $\text{u(t, a}{}_1\text{, …, a}{}_{\mathrm{n}}\text{)}\text{of}\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ defined in terms of the kernels k_i(t, s, a₁, …, a_n) in $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$ by is the identity operator on $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$, then the operator I ? K may be factored in the form (I ? K)(M ? W) = (I ? ΠK)(M ? ΠW). Here, W is an operator on $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ defined in terms of a kernel w(t, s, a₁, …, a_n) in $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$ by Wu = ∝_an^tw(t, s, a₁, …, a_n) u(s, a₁, …, a_n) ds. ΠW is the operator; ΠWu = ∝_{an ? 1}w(t, s, a₁, …, a_n) u(s, a₁, …, a_n) ds. ΠK is the operator; ΠKu = ∑_{i = 1}^{n ? 1} ∝_ai^tk_i(t, s, a₁, …, a_n) ds + ∝_{an ? 1}^tk_n(t, s, a₁, …, a_n) u(s, a₁, …, a_n) ds. The operator M is of the form m(t, a₁, …, a_n)I, where $\text{m ? A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ and maps elements of $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ into itself by multiplication. The function m is uniquely derived from K in the following manner. The operator K defines an operator $\text{K}{}^1$ on functions u in $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$, by . A determinant $\text{δ(I ? K}{}^1\text{)}$ of the operator $\text{I ? K}{}^1$ is defined as an element $\text{m}{}^1\text{(t, a}{}_1\text{, …, a}{}_{\mathrm{n\; +\; 1}}\text{)}$ of $\text{A(Ω}{}^{\mathrm{n\; +\; 2}}\text{)}$. This is mapped into $\text{A(Ω}{}^{\mathrm{n\; +\; 1}}\text{)}$ by setting a_{n + 1} = t to give m(t, a₁, …, a_n). The operator I ? ΠK may be factored in similar fashion, giving rise to a chain factorization of I ? K. In some cases all the matrix kernels k_i defining K are separable in the sense that k_i(t, s, a₁, …, a_n) = P_i(t, a₁, …, a_n) Q_i(s, a₁, …, a_n), where P_i is a 1 × p_i matrix and Q_i is a p_i × 1 matrix, each with elements in $\text{A(Ω}{}_{\mathrm{n\; +\; 1}}\text{)}$, explicit formulas are given for the kernels of the factors W. The various results are stated in a form allowing immediate extension to the vector-matrix case.     

On analytic functions whit gaps. II

Acta Mathematica Hungarica -     

Positive linear operators generated by analytic functions

Let φ be a power series with positive Taylor coefficients {a _k } _k=0^∞ and non-zero radius of convergence r ≤ ∞. Let ξ _x , 0 ≤ x < r be a random variable whose values α _k , k = 0, 1, …, are independent of x and taken with probabilities a _k x ^k /φ(x), k = 0, 1, …. The positive linear operator (A _φ f)(x):= E[f(ξ _x )] is studied. It is proved that if E(ξ _x ) = x, E(ξ _x ²) = qx ² + bx + c, q, b, c ∈ R, q > 0, then A _φ reduces to the Szász-Mirakyan operator in the case q = 1, to the limit q-Bernstein operator in the case 0 < q < 1, and to a modification of the Lupaş operator in the case q > 1.     

Superposition operators between weighted spaces of analytic functions

Abstract

We study the existence and the continuity of superposition operators between weighted spaces of holomorphic functions in terms of the weights.     

Weighted substitution operators in spaces of analytic functions

The spectrum of weighted conformai substitution operators is determined in a large class of Banach spaces of analytic functions, including many spaces whose norm is defined in terms of the properties of the functions themselves and of their derivatives.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 141, pp. 154–161, 1985.     

Multiplication operators on Hilbert spaces of analytic functions

Let be a Hilbert space of functions analytic on a plane domain such that for every in the functional of evaluation at is bounded. Assume further that contains the constants and admits multiplication by the independent variable z, M_z, as a bounded operator. We give sufficient conditions for M_z to be reflexive.Received: 17 February 2004     

解析函数空间上的自伴加权复合算子

通过再生核函数刻画了Hardy空间,Bergman空间上自伴加权复合算子以及自伴等距加权复合算子,最后研究了单位球上的分式线性自同构,得到了一个充分条件。     

Polynomial operators for spectral approximation of piecewise analytic functions

We propose the construction of a mixing filter for the detection of analytic singularities and an auto-adaptive spectral approximation of piecewise analytic functions, given either spectral or pseudo-spectral data, without knowing the location of the singularities beforehand. We define a polynomial frame with the following properties. At each point on the interval, the behavior of the coefficients in our frame expansion reflects the regularity of the function at that point. The corresponding approximation operators yield an exponentially decreasing rate of approximation in the vicinity of points of analyticity and a near best approximation on the whole interval. Unlike previously known results on the construction of localized polynomial kernels, we suggest a very simple idea to obtain exponentially localized kernels based on a general system of orthogonal polynomials, for which the Cesàro means of some order are uniformly bounded. The boundedness of these means is known in a number of cases, where no special function properties are known.     

Composition operators on a Banach space of analytic functions

In this paper, we study composition operators on a Banach space of analytic functions, denoted byX, which includes the Bloch space. This space arises naturally as the dual space of analytic functions in the Bergman spaceL _α ¹ (D) which admit an atomic decomposition. We characterize the functions which induce compact composition operators and those which induce Fredholm operatorson this space. We also investigate when a composition operator has a closed range. Supported by NNSFC No.19671036     

Iteration operators on a set of continuous functions of Baire space

Continuous functions on Baire space are considered. Iteration operators are defined on a set of continuous functions. The idea of a module of continuity of a function is introduced. The condition for the growth of module of continuity φ whose satisfaction guarantees that for any enumerable sequence of integration operators and any natural n there exists (n + 1) argument function with the module of continuity φ which cannot be obtained from n-argument functions with the module of continuity φ using any operator of this sequence is formulated. Examples of iteration operators are given.