Abstract
Let E be a compact subset of the complex plane with positive dx dy measure. For each p, 1 ⩽ p < ∞, let R p ( E dx dy) denote the closure in L p ( E, dx dy) of the rational functions having no poles on E. We give an example where E has empty interior and the functions in R p ( E, dx dy), p ⩾ 2, are uniquely determined by their values on any set of positive dx dy measure in E. Earlier, Sinanjan (Sibirsk Mat. Ž. 6 (1965), 1365–1381) obtained a somewhat weaker result along these lines and he also proved that this phenomenon can never occur when 1 ⩽ p < 2.
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