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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Bibliographic Details
Published in:Analysis and mathematical physics Vol. 16; no. 1; p. 2
Main Author: Biehler, Nikiforos
Format: Journal Article
Language:English
Published: Heidelberg Springer Nature B.V 01.02.2026
Subjects:
Approximation
Mathematical functions
Poles
Polynomials
ISSN:1664-2368, 1664-235X
Online Access:Get full text
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Summary: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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ISSN:1664-2368
1664-235X
DOI:10.1007/s13324-025-01145-8