Complex Chebyshev polynomials on circular sectors with degree six or less

Abstract

Let T n α T_n^\alpha denote the nth Chebyshev polynomial on the circular sector S α = { z : | z | ⩽ 1 , | arg ⁡ z | ⩽ α } {S^\alpha } = \{ z:|z| \leqslant 1,|\arg z| \leqslant \alpha \} . This paper contains numerical values of ‖ T n α ‖ ∞ {\left \| {T_n^\alpha } \right \|_\infty } and the corresponding coefficients of T n α T_n^\alpha for n = 1 ( 1 ) 6 n = 1(1)6 and α = 0 ∘ ( 5 ∘ ) 180 ∘ \alpha = {0^ \circ }({5^ \circ }){180^ \circ } . Also all critical angles for T n α , n = 1 ( 1 ) 6 T_n^\alpha ,n = 1(1)6 are listed, where an angle is called critical when the number of absolute maxima of | T n α | |T_n^\alpha | changes at that angle. All figures are given to six places. The positions (and hence the number) of extremal points of T n α , n = 1 ( 1 ) 6 T_n^\alpha ,n = 1(1)6 are presented graphically. The method consists of a combination of semi-infinite linear programming, finite linear programming, and Newton’s method.