scholarly journals On the rapidly convergence in capacity of the sequence of holomorphic functions

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2627-2633
Author(s):  
Kieu Chi
Keyword(s):  
Bounded Domain ◽  
Sufficient Conditions ◽  
Holomorphic Functions ◽  
Rational Functions ◽  
Borel Set ◽  
Constant Sequence ◽  
Bounded Holomorphic ◽  
Convergence In Capacity

In this paper, we are interested in finding sufficient conditions on a Borel set X lying either inside a bounded domain D ? Cn or in the boundary ?D so that if {rm}m?1 is a sequence of rational functions and {fm}m?1 is a sequence of bounded holomorphic functions on D with {fm-rm}m?1 is convergent fast enough to 0 in some sense on X then the convergence occurs on the whole domain D. The main result is strongly inspired by Theorem 3.6 in [3] whether the f fmg is a constant sequence.

scholarly journals Weighted Composition Operators fromH∞to the Bloch Space on the Polydisc

2007 ◽  
Vol 2007 ◽  
pp. 1-13 ◽  
Author(s):  
Songxiao Li ◽  
Stevo Stevic
Keyword(s):  
Holomorphic Function ◽  
Composition Operators ◽  
Bloch Space ◽  
Weighted Composition Operator ◽  
Sufficient Conditions ◽  
Holomorphic Functions ◽  
Weighted Composition Operators ◽  
Weighted Composition ◽  
Necessary And Sufficient ◽  
Bounded Holomorphic

LetDnbe the unit polydisc ofℂn,ϕ(z)=(ϕ1(z),…,ϕn(z))be a holomorphic self-map ofDn, andψ(z)a holomorphic function onDn. LetH(Dn)denote the space of all holomorphic functions with domainDn,H∞(Dn)the space of all bounded holomorphic functions onDn, andB(Dn)the Bloch space, that is,B(Dn)={f∈H(Dn)|‖f‖B=|f(0)|+supz∈Dn∑k=1n|(∂f/∂zk)(z)|(1−|zk|2)<+∞}. We give necessary and sufficient conditions for the weighted composition operatorψCϕinduced byϕ(z)andψ(z)to be bounded and compact fromH∞(Dn)to the Bloch spaceB(Dn).

WEIGHTED COMPOSITION OPERATORS BETWEEN H∞ AND GENERALLY WEIGHTED BLOCH SPACES ON POLYDISKS

2010 ◽  
Vol 21 (05) ◽  
pp. 687-699 ◽  
Author(s):  
HAIYING LI ◽  
PEIDE LIU
Keyword(s):  
Composition Operators ◽  
Bloch Space ◽  
Weighted Composition Operator ◽  
Sufficient Conditions ◽  
Holomorphic Functions ◽  
Weighted Composition Operators ◽  
Weighted Composition ◽  
Unit Polydisk ◽  
Necessary And Sufficient ◽  
Bounded Holomorphic

Let Un be the unit polydisk of Cn, φ(z) = (φ1(z),φ2(z),…,φn(z)) be a holomorphic self-map of Un and ψ be a holomorphic function on Un. H∞(Un) is the space of all bounded holomorphic functions on Un and by a generally weighted Bloch space we mean [Formula: see text]. We give necessary and sufficient conditions of the boundedness and compactness of the weighted composition operator ψCφ between H∞(Un) and [Formula: see text].

scholarly journals Slice holomorphic solutions of some directional differential equations with bounded L-index in the same direction

2019 ◽  
Vol 52 (1) ◽  
pp. 482-489 ◽  
Author(s):  
Andriy Bandura ◽  
Oleh Skaskiv ◽  
Liana Smolovyk
Keyword(s):  
Partial Differential Equations ◽  
Continuous Function ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Holomorphic Functions ◽  
Positive Continuous Function ◽  
Partial Differential ◽  
Partial Derivatives ◽  
Fixed Direction

AbstractIn the paper we investigate slice holomorphic functions F : ℂn → ℂ having bounded L-index in a direction, i.e. these functions are entire on every slice {z0 + tb : t ∈ℂ} for an arbitrary z0 ∈ℂn and for the fixed direction b ∈ℂn \ {0}, and (∃m0 ∈ ℤ+) (∀m ∈ ℤ+) (∀z ∈ ℂn) the following inequality holds{{\left| {\partial _{\bf{b}}^mF(z)} \right|} \over {m!{L^m}(z)}} \le \mathop {\max }\limits_{0 \le k \le {m_0}} {{\left| {\partial _{\bf{b}}^kF(z)} \right|} \over {k!{L^k}(z)}},where L : ℂn → ℝ+ is a positive continuous function, {\partial _{\bf{b}}}F(z) = {d \over {dt}}F\left( {z + t{\bf{b}}} \right){|_{t = 0}},\partial _{\bf{b}}^pF = {\partial _{\bf{b}}}\left( {\partial _{\bf{b}}^{p - 1}F} \right)for p ≥ 2. Also, we consider index boundedness in the direction of slice holomorphic solutions of some partial differential equations with partial derivatives in the same direction. There are established sufficient conditions providing the boundedness of L-index in the same direction for every slie holomorphic solutions of these equations.

scholarly journals The exterior Dirichlet problems of Monge–Ampère equations in dimension two

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Limei Dai
Keyword(s):  
Asymptotic Behavior ◽  
Unit Ball ◽  
Bounded Domain ◽  
Viscosity Solutions ◽  
Sufficient Conditions ◽  
Radial Solutions ◽  
Dirichlet Problems ◽  
Necessary And Sufficient ◽  
Monge Ampere ◽  
Prescribed Asymptotic Behavior

AbstractIn this paper, we study the Monge–Ampère equations $\det D^{2}u=f$ det D 2 u = f in dimension two with f being a perturbation of $f_{0}$ f 0 at infinity. First, we obtain the necessary and sufficient conditions for the existence of radial solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a unit ball. Then, using the Perron method, we get the existence of viscosity solutions with prescribed asymptotic behavior at infinity to Monge–Ampère equations outside a bounded domain.

DENSE STABLE RANK AND RUNGE-TYPE APPROXIMATION THEOREMS FOR MAPS

2021 ◽  
pp. 1-29
Author(s):  
ALEXANDER BRUDNYI
Keyword(s):  
Disjoint Union ◽  
A Priori ◽  
Holomorphic Functions ◽  
Stable Rank ◽  
Open Unit ◽  
Uniform Estimates ◽  
Approximation Theorems ◽  
Type Approximation ◽  
Similar Approximation ◽  
Bounded Holomorphic

Abstract Let $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ be the Banach algebra of bounded holomorphic functions defined on the disjoint union of countably many copies of the open unit disk ${\mathbb {D}}\subset {{\mathbb C}}$ . We show that the dense stable rank of $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ is $1$ and, using this fact, prove some nonlinear Runge-type approximation theorems for $H^\infty ({\mathbb {D}}\times {\mathbb {N}})$ maps. Then we apply these results to obtain a priori uniform estimates of norms of approximating maps in similar approximation problems for the algebra $H^\infty ({\mathbb {D}})$ .

scholarly journals A note on a space Hp, a of holomorphic functions

1987 ◽  
Vol 35 (3) ◽  
pp. 471-479
Author(s):  
H. O. Kim ◽  
S. M. Kim ◽  
E. G. Kwon
Keyword(s):  
Hardy Space ◽  
Complex Plane ◽  
Unit Disc ◽  
Weak Type ◽  
Holomorphic Functions ◽  
Embedding Theorem ◽  
Type Operator ◽  
Taylor Coefficients ◽  
Bounded Holomorphic

For 0 < p < ∞ and 0 ≤a; ≤ 1, we define a space Hp, a of holomorphic functions on the unit disc of the complex plane, for which Hp, 0 = H∞, the space of all bounded holomorphic functions, and Hp, 1 = Hp, the usual Hardy space. We introduce a weak type operator whose boundedness extends the well-known Hardy-Littlewood embedding theorem to Hp, a, give some results on the Taylor coefficients of the functions of Hp, a and show by an example that the inner factor cannot be divisible in Hp, a.

scholarly journals Lagrangian controllability of inviscid incompressible fluids: a constructive approach

2017 ◽  
Vol 23 (3) ◽  
pp. 1179-1200
Author(s):  
Thierry Horsin ◽  
Otared Kavian
Keyword(s):  
Numerical Simulations ◽  
Euler Equation ◽  
Approximate Controllability ◽  
Holomorphic Functions ◽  
Rational Functions ◽  
Incompressible Fluids ◽  
Constructive Method ◽  
Constructive Approach ◽  
Density Result

We present here a constructive method of Lagrangian approximate controllability for the Euler equation. We emphasize on different options that could be used for numerical recipes: either, in the case of a bi-dimensionnal fluid, the use of formal computations in the framework of explicit Runge approximations of holomorphic functions by rational functions, or an approach based on the study of the range of an operator by showing a density result. For this last insight in view of numerical simulations in progress, we analyze through a simplified problem the observed instabilities.

Angular and Tangential Limits of Blaschke Products and their Successive Derivatives

1962 ◽  
Vol 14 ◽  
pp. 334-348 ◽  
Author(s):  
G. T. Cargo
Keyword(s):  
Blaschke Product ◽  
Common Knowledge ◽  
Holomorphic Functions ◽  
Blaschke Products ◽  
Radial Limit ◽  
Radial Limits ◽  
Almost Everywhere ◽  
Image Position ◽  
Bounded Holomorphic ◽  
Tangential Limits

In this paper, we shall be concerned with bounded, holomorphic functions of the formwhere(1)(2)and(3)B(z{an}) is called a Blaschke product, and any sequence {an} which satisfies (2) and (3) is called a Blaschke sequence. For a general discussion of the properties of Blaschke products, see (18, pp. 271-285) or (14, pp. 49-52).According to a theorem due to Riesz (15), a Blaschke product has radial limits of modulus one almost everywhere on C = {z: |z| = 1}. Moreover, it is common knowledge that, if a Blaschke product has a radial limit at a point, then it also has an angular limit at the point (see 14, p. 19 and 6, p. 457).

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