Uniqueness of solutions to a twodimensional mean problem
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Let 0 < rm < 1, rmm ≤ ρ{variant} for all large m, and let wn = e i2π n, n = 1, 2, 4 For a function f{hook}(z) = ∑ anzn, holomorphic in the open unit disk U, let sn(f{hook}) = ( 1 n) ∑k = 1nf{hook}(rnwnk), the nth arithmetic mean of f{hook} over the circle z = rn. We prove that if p < 1 and an = O(nα1) for α1 = 1.728..., then f{hook} is uniquely determined by the twodimensional means sn(f{hook}), n = 1, 2, 4 We also prove that for each ρ{variant}, 0 < ρ{variant} < 1, there is a nontrivial f{hook}, holomorphic in U. such that sn(f{hook}) = 0 for n = 1, 2,... with rn = ρ{variant} 1 n. © 1978.
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Borosh, I., & Chui, C. K.
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