Browsing by Subject "Bergman projections"

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    Extremal problems and bergman projections
    (2017-07) Özbek, Rasimcan
    Studying extremal problems on Bergman spaces is rather new and techniques used are usually specific to the problem to be solved. However, a 2014 paper by T. Ferguson developed a systematic method using Bergman projections for solving extremal problems on Bergman spaces Ap on the unit disc with 1 < p < 1. We extended this method to weighted Bergman spaces Ap and in some special cases, to extremal problems defined by linear functionals of evaluations at points in the disc other than the origin. We computed the kernels of several such functionals. We also computed the Bergman projections of some functions related to these kernels. Using these projections, we solved a few extremal problems explicitly. Our results have the potential to be extended to the case p = 1 and to more general spaces. We also gave a proof of the existence of the solutions to the extremal problems on the Bergman spaces A1 defined by functionals with kernels that extend to the closed disc continuously.
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    Extremal problems on Bergman spaces A¹α and Besov spaces
    (2022-05) Balcı, Alper
    Extremal problems in different function spaces have long been investigated. Ferguson provides a method, using Bergman projections, to solve certain types of extremal problems in Bergman spaces for 1 < p < ∞ in his work [3]. Later the method is extended to weighted Bergman spaces for 1 < p < ∞ in [13]. Now, we extend this method to the p = 1 case. The two cases differ in the structure of Bergman projections and dual spaces. First, we define some function spaces, namely weighted Bergman spaces, the Bloch space, and Besov spaces, and show the usage of Bergman projection on these spaces. Then, we find some conditions to ensure the existence of unique solutions for extremal problems. Later, we use Bergman projection to find a candidate function for the solution in the p = 1 case, and we prove that the candidate function is the solution if it never attains the value 0. Finally, under special conditions, we solve a similar problem in Besov spaces.