TY - JOUR
T1 - Chebyshev polynomials corresponding to a vanishing weight
AU - Bergman, Alex
AU - Rubin, Olof
PY - 2024/5/2
Y1 - 2024/5/2
N2 - We consider weighted Chebyshev polynomials on the unit circle corresponding to a weight of the
form (z−1)s where s > 0. For integer values of s this corresponds to prescribing a zero of the polynomial
on the boundary. As such, we extend findings of Lachance et al. (1979), to non-integer s. Using this
generalisation, we are able to relate Chebyshev polynomials on lemniscates and other, more established,
categories of Chebyshev polynomials. An essential part of our proof involves the broadening of the
Erdos–Lax inequality to encompass powers of polynomials. We believe that this particular result holds ˝
significance in its own right.
AB - We consider weighted Chebyshev polynomials on the unit circle corresponding to a weight of the
form (z−1)s where s > 0. For integer values of s this corresponds to prescribing a zero of the polynomial
on the boundary. As such, we extend findings of Lachance et al. (1979), to non-integer s. Using this
generalisation, we are able to relate Chebyshev polynomials on lemniscates and other, more established,
categories of Chebyshev polynomials. An essential part of our proof involves the broadening of the
Erdos–Lax inequality to encompass powers of polynomials. We believe that this particular result holds ˝
significance in its own right.
U2 - 10.1016/j.jat.2024.106048
DO - 10.1016/j.jat.2024.106048
M3 - Article
SN - 0021-9045
VL - 301
JO - Journal of Approximation Theory
JF - Journal of Approximation Theory
M1 - 106048
ER -