Cesàro Sum Approximation of Outer Functions

نویسندگان

  • R. W. Barnard
  • K. Pearce
  • J. Cima
چکیده

It is well known that outer functions are zero-free on the unit disk. If an outer function, f , is given as an infinite series and a finite (polynomial) approximation is chosen, then it is desirable that the approximants retain the zero-free property of f . We observe for outer functions that the standard Taylor approximants do not, in general, retain the zero-free property – even when fairly restrictive conditions are placed on the permissible outer functions. We show, using methods of geometric function theory, that Cesàro sum approximants for outer functions which arise as the derivatives of bounded convex functions do inherit the desired zero-free property . We, also, find that a “cone-like” condition holds for the boundaries of the ranges of these approximants.

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تاریخ انتشار 1998