Extreme Points of the Unit Ball B4 in a Space of Real Polynomials of Degree at most Four with the Supremum Norm
Sprache des Vortragstitels:
Englisch
Original Tagungtitel:
SYNASC
Sprache des Tagungstitel:
Englisch
Original Kurzfassung:
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].
Forschungs-