Extremal functions in weighted Bergman spaces

Abstract

We find an explicit formula for the solution of an extremal problem for the invariant subspaces in the weighted Bergman spaces , , consisting of functions having one zero of order at least k, . A consequence of this formula is the failure of the Beurling-type theorem for . We also consider the analogous extremal problem for , .

Keywords:

• Weighted Bergman spaces
• invariant subspaces
• Beurling-type theorem

AMS Subject Classifications:

• 30H20
• 47A15

Acknowledgements

The authors are very grateful to the anonymous reviewer for careful reading of the manuscript and helpful comments.

Notes

No potential conflict of interest was reported by the authors.