TY - JOUR
T1 - Approximation in the mean and quasianalyticity
AU - Brennan, James E.
PY - 1973/3
Y1 - 1973/3
N2 - Let E be a compact subset of the complex plane with positive dx dy measure. For each p, 1 ≤ p < ∞, let Rp(E dx dy) denote the closure in Lp(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 Rp(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.
AB - Let E be a compact subset of the complex plane with positive dx dy measure. For each p, 1 ≤ p < ∞, let Rp(E dx dy) denote the closure in Lp(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 Rp(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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U2 - 10.1016/0022-1236(73)90082-7
DO - 10.1016/0022-1236(73)90082-7
M3 - Article
AN - SCOPUS:0007467770
SN - 0022-1236
VL - 12
SP - 307
EP - 320
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 3
ER -