Complex Approximation and Simultaneous Interpolation on Closed Sets

1977 ◽  
Vol 29 (4) ◽  
pp. 701-706 ◽  
Author(s):  
P. M. Gauthier ◽  
W. Hengartner
Keyword(s):  
Analytic Function ◽  
Complex Plane ◽  
Closed Subset ◽  
Finite Complex ◽  
Complex Approximation ◽  
Closed Sets ◽  
Limit Points ◽  
Simultaneous Interpolation ◽  
Complex Valued

Let ƒ be a complex-valued function denned on a closed subset F of the finite complex plane C, and let {Zn} be a sequence on F without limit points. We wish to find an analytic function g which simultaneously approximates ƒ uniformly on F and interpolates ƒ at the points {Zn}.

Lip α Approximation on Closed Sets with Lip α Extension

1995 ◽  
Vol 38 (1) ◽  
pp. 23-33
Author(s):  
A. Bonilla ◽  
J. C. Fariña
Keyword(s):  
Complex Plane ◽  
Closed Subset ◽  
Closed Sets

AbstractLet F be a relatively closed subset of a domain G in the complex plane. Let f be a function that is the limit, in the Lip α norm on F, of functions which are holomorphic or meromorphic on G (0 < α < 1). We prove that, under the same conditions that give Lip α-approximation (0 < α < 1 ) on relatively closed subsets of G, it is possible to choose the approximating function m in such a way that f — m can be extended to a function belonging to lip

scholarly journals Pick’s theorems for dissipative operators

Filomat ◽  
2016 ◽  
Vol 30 (6) ◽  
pp. 1591-1599
Author(s):  
H.M. Srivastava ◽  
A.K. Mishra
Keyword(s):  
Hilbert Space ◽  
Analytic Function ◽  
Complex Plane ◽  
Complex Hilbert Space ◽  
Dissipative Operator ◽  
Bounded Linear ◽  
Dissipative Operators ◽  
Proper Contraction ◽  
Complex Valued ◽  
Bounded Linear Transformation

Let H be a complex Hilbert space and let A be a bounded linear transformation on H. For a complex-valued function f, which is analytic in a domain D of the complex plane containing the spectrum of A, let f (A) denote the operator on H defined by means of the Riesz-Dunford integral. In the present paper, several (presumably new) versions of Pick?s theorems are proved for f (A), where A is a dissipative operator (or a proper contraction) and f is a suitable analytic function in the domain D.

Uniform And Tangential Approximations By Meromorphic Functions on Closed Sets

1976 ◽  
Vol 28 (1) ◽  
pp. 104-111 ◽  
Author(s):  
Alice Roth
Keyword(s):  
Complex Plane ◽  
Meromorphic Functions ◽  
Rational Functions ◽  
Proper Subset ◽  
Compact Set ◽  
Finite Complex ◽  
Open Domain ◽  
Closed Sets

Let G be an (open) domain in the finite complex plane and F a relatively closed proper subset of G. We denote by M(G) the set of functions meromorphic on G and as usual by R(K) (for a compact set K) the set of uniform limits of rational functions without poles on K.

On BMOA for Riemann Surfaces

1981 ◽  
Vol 33 (5) ◽  
pp. 1255-1260 ◽  
Author(s):  
Thomas A. Metzger
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Complex Plane ◽  
Riemann Surfaces ◽  
Hardy Class ◽  
Image Position

Let Δ denote the unit disk in the complex plane C. The space BMO has been extensively studied by many authors (see [3] for a good exposition of this topic). Recently, the subspace BMOA (Δ) has become a topic of interest. An analytic function f, in the Hardy class H2(A), belongs to BMOA (Δ) if(1)whereIt is known (see [3, p. 96]) that (1) is equivalent to(2)

Asymptotic Values Along Julia Rays

1976 ◽  
Vol 28 (6) ◽  
pp. 1210-1215
Author(s):  
P. M. Gauthier ◽  
J. S. Hwang
Keyword(s):  
Complex Plane ◽  
Finite Complex ◽  
Asymptotic Values ◽  
The Family ◽  
Definition Of

Let ƒ be a function meromorphic in the finite complex plane C. If for some number θ, 0 ≦ θ < 2 π, the family, fr(z) = f(reθz), is not normal at z = 1, then the ray arg z = θ is called a Julia ray. Such a ray has the property that in every sector containing it, F assumes every value infinitely often with at most two exceptions. Many authors have taken this property as the definition of a Julia ray.

Possibility of Uniform Rational Approximation in the Spherical Metric

1976 ◽  
Vol 28 (1) ◽  
pp. 112-115 ◽  
Author(s):  
P. M. Gauthier ◽  
A. Roth ◽  
J. L. Walsh
Keyword(s):  
Compact Subset ◽  
Rational Approximation ◽  
Complex Plane ◽  
Riemann Sphere ◽  
Rational Functions ◽  
Finite Complex ◽  
Finite Plane ◽  
Extended Plane ◽  
Chordal Metric

Let ƒ b e a mapping defined on a compact subset K of the finite complex plane C and taking its values on the extended plane C ⋃ ﹛ ∞﹜. For x a metric on the extended plane, we consider the possibility of approximating ƒ x-uniformly on K by rational functions. Since all metrics on C ⋃ ﹛oo ﹜ are equivalent, we shall consider that x is the chordal metric on the Riemann sphere of diameter one resting on a finite plane at the origin.

scholarly journals Tangent vectors to sets in the theory of geodesics

1987 ◽  
Vol 106 ◽  
pp. 29-47 ◽  
Author(s):  
Dumitru Motreanu
Keyword(s):  
Closed Subset ◽  
Tangent Vector ◽  
General Concept ◽  
Recent Book ◽  
Closed Sets ◽  
Banach Manifolds ◽  
Flow Invariance ◽  
Tangent Vectors

In the setting of Banach manifolds the notion of tangent vector to an arbitrary closed subset has been introduced in [11] by the author and N. H. Pavel, and it has been used in flow-invariance and optimization ([11], [12], [13]). For detailed informations on tangent vectors to closed sets (including historical comments) we refer to the recent book of N. H. Pavel [17].The aim of this paper is to apply this general concept of tangency in the study of geodesies. We are concerned with geodesies which have either the endpoints in given closed subsets or the same angle for a fixed closed subset. This approach allows one to extend important results due to K. Grove [4] and T. Kurogi ([6], [7]).

scholarly journals Approximate Closed-Form Formulas for the Zeros of the Bessel Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Rafael G. Campos ◽  
Marisol L. Calderón
Keyword(s):  
Closed Form ◽  
Complex Plane ◽  
Complex Number ◽  
Bessel Polynomials ◽  
The Real ◽  
Bessel Polynomial ◽  
Limit Points ◽  
Electrostatic Interpretation ◽  
Approximate Expressions

We find approximate expressionsx̃(k,n,a)andỹ(k,n,a)for the real and imaginary parts of thekth zerozk=xk+iykof the Bessel polynomialyn(x;a). To obtain these closed-form formulas we use the fact that the points of well-defined curves in the complex plane are limit points of the zeros of the normalized Bessel polynomials. Thus, these zeros are first computed numerically through an implementation of the electrostatic interpretation formulas and then, a fit to the real and imaginary parts as functions ofk,nandais obtained. It is shown that the resulting complex numberx̃(k,n,a)+iỹ(k,n,a)isO(1/n2)-convergent tozkfor fixedk.

scholarly journals The problem of missing terms in term by term integration involving divergent integrals

2017 ◽  
Vol 473 (2197) ◽  
pp. 20160567 ◽  
Author(s):  
Eric A. Galapon
Keyword(s):  
Analytic Continuation ◽  
Complex Plane ◽  
Finite Part ◽  
Stieltjes Transform ◽  
Complex Valued

Term by term integration may lead to divergent integrals, and naive evaluation of them by means of, say, analytic continuation or by regularization or by the finite part integral may lead to missing terms. Here, under certain analyticity conditions, the problem of missing terms for the incomplete Stieltjes transform, ∫ 0 a f ( x ) ( ω + x ) − 1   d x , and the Stieltjes transform itself, ∫ 0 ∞ f ( x ) ( ω + x ) − 1   d x , is resolved by lifting the integration in the complex plane. It is shown that the missing terms arise from the singularities of the complex-valued function f ( z )( ω + z ) −1 , with the divergent integrals arising from term by term integration interpreted as finite part integrals.

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