Geodesic
Unconditional Exponential Bases in Bergman Spaces
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Complex analysis and applications, Tome 253 (2006), pp. 88-100

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It is proved that unconditional exponential bases cannot be constructed in the Bergman space $B_2(D)$ in the case when $D$ is a bounded convex domain in the plane such that at least at one point of its boundary the curvature exists and is different from zero. 
@article{TM_2006_253_a7,
     author = {K. P. Isaev and R. S. Yulmukhametov},
     title = {Unconditional {Exponential} {Bases} in {Bergman} {Spaces}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {88--100},
     publisher = {mathdoc},
     volume = {253},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2006_253_a7/}
}
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K. P. Isaev; R. S. Yulmukhametov. Unconditional Exponential Bases in Bergman Spaces. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Complex analysis and applications, Tome 253 (2006), pp. 88-100. http://geodesic.mathdoc.fr/item/TM_2006_253_a7/