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VIDEO LIBRARY |
The 27th International Conference on Finite and Infinite Dimensional Complex Analysis and Applications
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On Chui's conjecture and approximation by simplest fractions K. Yu. Fedorovskiyab a Bauman Moscow State Technical University b St. Petersburg State University, Mathematics and Mechanics Faculty |
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Abstract: In 1971 C. K. Chui conjectured that the average field strength in the unit disk $$ \bigg\|\sum_{k=1}^{N}\frac1{z-z_k}\bigg\|_{L^1(\mathbb D)}\geqslant \bigg\|\sum_{k=1}^{N}\frac1{z-\omega_N^k}\bigg\|_{L^1(\mathbb D)}, $$ where This conjecture remains open, and in the talk we will consider its analogue for the weighted Bergman spaces $$ \|f\|_{2,\alpha}^2:=\frac{\alpha+1}{\pi}\int|f(z)|^2(1-|z|^2)^\alpha\,dxdy<\infty. $$ It will be shown that the statement analogous to Chui's conjecture is true for the spaces $$ \bigg\|\sum_{k=1}^{N}\frac1{z-z_k}\bigg\|_{2,\alpha}\geqslant \bigg\|\sum_{k=1}^{N}\frac1{z-\omega_N^k}\bigg\|_{2,\alpha}. $$ It is planned to consider also the problem about completeness in the space $$ \sum_{k=1}^{N}\frac1{z-z_k}, $$ where This is a joint work with E. Abakumov \textup(University Paris Est, Marne-la-Vallée, France\textup) and A. Borichev \textup(Aix–Marseille University, France\textup). Language: English |