Zeros of optimal polynomial approximants in ℓ^p_A [article]

Raymond Cheng, William T. Ross, Daniel Seco
2021 arXiv   pre-print
The study of inner and cyclic functions in ℓ^p_A spaces requires a better understanding of the zeros of the so-called optimal polynomial approximants. We determine that a point of the complex plane is the zero of an optimal polynomial approximant for some element of ℓ^p_A if and only if it lies outside of a closed disk (centered at the origin) of a particular radius which depends on the value of p. We find the value of this radius for p≠ 2. In addition, for each positive integer d there is a
more » ... ynomial f_d of degree at most d that minimizes the modulus of the root of its optimal linear polynomial approximant. We develop a method for finding these extremal functions f_d and discuss their properties. The method involves the Lagrange multiplier method and a resulting dynamical system.
arXiv:2104.08014v1 fatcat:2jdwfnc6erhipeddcrfd25grp4