Extreme Points of the Unit Ball B4 in a Space of Real Polynomials of Degree at most Four with the Supremum Norm

Description

Let p 2 Pn and let N(p) be the number of all zeros of the polynomial 1 − p2 in the interval I = [−1, 1], counted with according multiplicity. In the case of real polynomials, Konheim and Rivlin in [4] proved that p 2 EBn if and only if N(p) > n. We will exploit this fact heavily in our work. We know that EB0 = {−1, 1} and EB1 = {−1, 1,−x, x}. If n > 1, there are infinitely many polynomials present in EBn, so their explicit description is a difficult task. Szumny [8] showed that quadratic polynomials in EB2 are just the quadratic improper Zolotarev polynomials. Moreover, Sok´ol and Szumny [7] showed that the cubic polynomials of EB3 consist of 2 one-parameter and 1 two-parameter polynomial family and remarked that the structure of EB4 seems to be “very complicated”. In [9] the same authors considered only those element from EB4 for which N(p) > 5. Our goal now is to give a complete description of EB4. For the related geometric investigation of the space of real polynomials equipped with the L1 norm, we refer the interested reader to [2] and for the space of complex polynomials equipped with the sup norm to [3].

Period	20 Sept 2022
Event title	SYNASC
Event type	Conference
Location	AustriaShow on map