High degree simple partial fractions in the Bergman space: Approximation and Optimization
We consider the class of standard weighted Bergman spaces Aα2(D) and the set SFN(T) of simple partial fractions of degree N with poles on the unit circle. We prove that under certain conditions, the simple partial fractions of order N, with n poles on the unit circle attain minimal norm if and only...
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| Veröffentlicht in: | Analysis and mathematical physics Jg. 16; H. 1; S. 2 |
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01.02.2026
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| Abstract | We consider the class of standard weighted Bergman spaces Aα2(D) and the set SFN(T) of simple partial fractions of degree N with poles on the unit circle. We prove that under certain conditions, the simple partial fractions of order N, with n poles on the unit circle attain minimal norm if and only if the points are equidistributed on the unit circle. We show that this is not the case if the conditions we impose are not met, exhibiting a new interesting phenomenon. We find sharp asymptotics for these norms. Additionally we describe the closure of these fractions in the standard weighted Bergman spaces. |
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| AbstractList | We consider the class of standard weighted Bergman spaces Aα2(D) and the set SFN(T) of simple partial fractions of degree N with poles on the unit circle. We prove that under certain conditions, the simple partial fractions of order N, with n poles on the unit circle attain minimal norm if and only if the points are equidistributed on the unit circle. We show that this is not the case if the conditions we impose are not met, exhibiting a new interesting phenomenon. We find sharp asymptotics for these norms. Additionally we describe the closure of these fractions in the standard weighted Bergman spaces. |
| ArticleNumber | 2 |
| Author | Biehler, Nikiforos |
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| Cites_doi | 10.1090/s0002-9947-1969-0237795-3 10.1007/s10958-011-0440-5 10.1007/s00208-020-02114-1 10.1070/im2014v078n06abeh002721 10.1080/00029890.1971.11992851 10.1080/00029890.1972.11993174 10.1070/sm9298 10.1134/S0001434621010065 10.1070/sm1995v082n02abeh003573 10.1070/sm8500 10.1090/s0002-9939-1971-0277730-x 10.2307/1970655 10.1007/978-1-4612-0497-8 10.1215/s0012-7094-49-01644-0 |
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| References | DJ Newman (1145_CR14) 1972; 79 M Thompson (1145_CR16) 1967; 159 E Abakumov (1145_CR1) 2021; 379 PA Borodin (1145_CR4) 2014; 78 H Hedenmalm (1145_CR11) 2000 CK-T Chui (1145_CR7) 1969; 138 CK-T Chui (1145_CR6) 1971; 78 NA Dyuzhina (1145_CR10) 2021; 109 1145_CR13 VI Danchenko (1145_CR9) 2011; 176 VI Danchenko (1145_CR8) 1995; 82 NK Bary (1145_CR2) 2014 Z Rubinstein (1145_CR15) 1971; 29 J Korevaar (1145_CR12) 1964; 80 PA Borodin (1145_CR5) 2021; 212 PA Borodin (1145_CR3) 2016; 207 |
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| Title | High degree simple partial fractions in the Bergman space: Approximation and Optimization |
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