## Abstract

In this paper we study the spaces of continuous functions f on ℝ^{2} satisfying {Mathematical expression} uniformly on compact sets as ξ → ∞, where μ is a finite complex Borel measure supported on the unit circle {z ∈ ℂ: |z|=1} with ∫^{2π}_{0} dμ(θ)=1. Our main result is the following: Theorem. Let μ be a finite Borel measure on the unit circle in ℝ with ∫^{2π}_{0} dμ(θ)=1. Let I=∩ (a_{n} b_{n}) be a union of disjoint intervals in ℝ^{+} so that lim sup(b_{n} - a_{n}) > 0. Then: (a) Every f e{open} C(ℝ^{2}) satisfying {Mathematical expression} uniformly on compact sets, as ξ → ∞, |ξ| e{open} I, is harmonic. If {Mathematical expression}(I) ≠ 0 then f is analytic. (b) If {Mathematical expression}(k)=0, k=-1, -2,..., -N, {Mathematical expression}(- N-1) ≠ 0 and {Mathematical expression}(I) ≠ 0 [ {Mathematical expression}(k)=0, k=1, 2,..., N, {Mathematical expression}(N+1) ≠ 0 and {Mathematical expression}(-1) ≠ 0] then f e{open} C(ℝ^{2}) satisfies (4) if and only if f is a polynomial in z [in {Mathematical expression}] of degree not exceeding N. Hence, if {Mathematical expression}(k)=0 for k=±1, ±2, ..., ± N and {Mathematical expression}(±(N+1)) ≠ 0, (4) characterizes the set of harmonic polynomials of degree not exceeding N. (c) If μ is analytic (conjugate-analytic) with {Mathematical expression}(1) ≠ 0 ( {Mathematical expression}(-1) ≠ 0) then f e{open} C(ℝ^{2}) satisfies (4) if and only if f is analytic (conjugate-analytic).

Original language | English |
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Pages (from-to) | 242-247 |

Number of pages | 6 |

Journal | Aequationes Mathematicae |

Volume | 41 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1991 |

## Keywords

- AMS (1980) subject classification: Primary 30E15, 26C05, 30C10

## ASJC Scopus subject areas

- General Mathematics
- Discrete Mathematics and Combinatorics
- Applied Mathematics