On a generalized asymptotic mean value property

In this paper we study the spaces of continuous functions f on ℝ² 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 ∫^2π₀ dμ(θ)=1. Our main result is the following: Theorem. Let μ be a finite Borel measure on the unit circle in ℝ with ∫^2π₀ 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(ℝ²) 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(ℝ²) 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(ℝ²) satisfies (4) if and only if f is analytic (conjugate-analytic).