# On a class of holomorphic functions representable by Carleman formulas in the interior of an equilateral cone from their values on its rigid base

George Chailos, Alekos Vidras

Research output: Contribution to journalArticle

3 Citations (Scopus)

### Abstract

Let Δ be an equilateral cone in C with vertices at the complex numbers 0, z1 0, z2 0 and rigid base M (Section 1). Assume that the positive real semi-axis is the bisectrix of the angle at the origin. For the base M of the cone Δ we derive a Carleman formula representing all those holomorphic functions f ∈ H (Δ) from their boundary values (if they exist) on M which belong to the class NHM 1 (Δ). The class NHM 1 (Δ) is the class of holomorphic functions in Δ which belong to the Hardy class H1 near the base M (Section 2). As an application of the above characterization, an important result is an extension theorem for a function f ∈ L1 (M) to a function f ∈ NHM 1 (Δ).

Original language English 657-672 16 Journal of Mathematical Analysis and Applications 310 2 https://doi.org/10.1016/j.jmaa.2005.02.036 Published - 15 Oct 2005

### Fingerprint

Equilateral
Cones
Analytic function
Interior
Cone
Hardy Class
Extension Theorem
Complex number
Boundary Value
Angle
Class

### Keywords

• Carleman formula
• Cone with a rigid base

### Cite this

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title = "On a class of holomorphic functions representable by Carleman formulas in the interior of an equilateral cone from their values on its rigid base",
abstract = "Let Δ be an equilateral cone in C with vertices at the complex numbers 0, z1 0, z2 0 and rigid base M (Section 1). Assume that the positive real semi-axis is the bisectrix of the angle at the origin. For the base M of the cone Δ we derive a Carleman formula representing all those holomorphic functions f ∈ H (Δ) from their boundary values (if they exist) on M which belong to the class NHM 1 (Δ). The class NHM 1 (Δ) is the class of holomorphic functions in Δ which belong to the Hardy class H1 near the base M (Section 2). As an application of the above characterization, an important result is an extension theorem for a function f ∈ L1 (M) to a function f ∈ NHM 1 (Δ).",
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In: Journal of Mathematical Analysis and Applications, Vol. 310, No. 2, 15.10.2005, p. 657-672.

Research output: Contribution to journalArticle

TY - JOUR

T1 - On a class of holomorphic functions representable by Carleman formulas in the interior of an equilateral cone from their values on its rigid base

AU - Chailos, George

AU - Vidras, Alekos

PY - 2005/10/15

Y1 - 2005/10/15

N2 - Let Δ be an equilateral cone in C with vertices at the complex numbers 0, z1 0, z2 0 and rigid base M (Section 1). Assume that the positive real semi-axis is the bisectrix of the angle at the origin. For the base M of the cone Δ we derive a Carleman formula representing all those holomorphic functions f ∈ H (Δ) from their boundary values (if they exist) on M which belong to the class NHM 1 (Δ). The class NHM 1 (Δ) is the class of holomorphic functions in Δ which belong to the Hardy class H1 near the base M (Section 2). As an application of the above characterization, an important result is an extension theorem for a function f ∈ L1 (M) to a function f ∈ NHM 1 (Δ).

AB - Let Δ be an equilateral cone in C with vertices at the complex numbers 0, z1 0, z2 0 and rigid base M (Section 1). Assume that the positive real semi-axis is the bisectrix of the angle at the origin. For the base M of the cone Δ we derive a Carleman formula representing all those holomorphic functions f ∈ H (Δ) from their boundary values (if they exist) on M which belong to the class NHM 1 (Δ). The class NHM 1 (Δ) is the class of holomorphic functions in Δ which belong to the Hardy class H1 near the base M (Section 2). As an application of the above characterization, an important result is an extension theorem for a function f ∈ L1 (M) to a function f ∈ NHM 1 (Δ).

KW - Carleman formula

KW - Cone with a rigid base

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