Cyclicity in Dirichlet-type spaces and extremal polynomials Articles uri icon

publication date

  • April 2015

start page

  • 259

end page

  • 286

issue

  • 1

volume

  • 126

International Standard Serial Number (ISSN)

  • 0021-7670

Electronic International Standard Serial Number (EISSN)

  • 1565-8538

abstract

  • For functions f in Dirichlet-type spaces Dalfa, we study how to determine constructively optimal polynomials p n that minimize ∥pf−1∥alfa among all polynomials p of degree at most n. We then obtain sharp estimates for the rate of decay of ∥pnf−1∥alfa as n approaches ∞, for certain classes of functions f. Finally, inspired by the Brown-Shields conjecture, we prove that certain logarithmic conditions on f imply cyclicity, and we study some computational phenomena pertaining to the zeros of optimal polynomials.

keywords

  • bergman space; optimal norm; open unit disk; dirichlet space; closed disk