Geodesic
The absence of unconditional bases of exponentials in Bergman spaces on non-polygonal domains
Izvestiya. Mathematics , Tome 71 (2007) no. 6, pp. 1145-1166.

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We prove that it is impossible to construct unconditional bases of exponentials in the Bergman space $B_2(D)$ in the case when $D$ is a bounded convex domain on the plane such that at some point of the boundary the curvature exists and is different from zero. 
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K. P. Isaev; R. S. Yulmukhametov. The absence of unconditional bases of exponentials in Bergman spaces on non-polygonal domains. Izvestiya. Mathematics , Tome 71 (2007) no. 6, pp. 1145-1166. http://geodesic.mathdoc.fr/item/IM2_2007_71_6_a3/

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