Extremal Problems for Nonvanishing Functions in Bergman Spaces
Dov Aharonov2005, Operator Theory: Advances and Applications
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Abstract
In this paper, we study general extremal problems for non-vanishing functions in Bergman spaces. We show the existence and uniqueness of solutions to a wide class of such problems. In addition, we prove certain regularity results: the extremal functions in the problems considered must be in a Hardy space, and in fact must be bounded. We conjecture what the exact form of the extremal function is. Finally, we discuss the specific problem of minimizing the norm of non-vanishing Bergman functions whose first two Taylor coefficients are given.
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with whom I have shared the office. I am also extremely grateful to Josep Vives from the Department of Probability, Logic and Statistics. All of them were available when I wanted to talk maths as well as those times when I would have preferred to speak about anything else. To my close friends and family I will be eternally grateful. Without your encouragement I would never have completed this work. You will never know how much your love, kindness and support has meant to me these past years and I just hope that I can pay it all back in the years to come. v vi
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