Tchebychev Polynomials of Second Kind on the Ellipse and Approximations

We study the orthogonality of Tchebychev polynomials of second kind {Un (z)}n=0,1,2,3,...with respect to the Lebesgue planar measure concentrated on the ellipse D: b2x2 + a2y2 < a2b2 where a > b,a system of orthogonal polynomials, given by : \(U_n(z)={Tₙ₊₁^{}(z)}{n+1}={sin ((n+1) cos ⁻¹ z)}{√1-z^2},\) n = 0, 1, 2, 3, ... where Tn (z) = cos ( n cos−1 z ) , n = 0, 1, 2, 3, ...is a polynomial of degree n . Tn (z) is called the Tchebychev polynomial of degree n of first kind.They satisfies \(_D U_n(z) U_m(z)̄ d x d y={4(n+1)}{π(ρⁿ⁺¹-ρ⁻ⁿ⁻¹)} δn, m\) , n,m = 0, 1, 2, 3, ... where δn,m , is the symbol of Kronecker and (a + b)2 = ρ. We study extremal properties and minimization and Fourier development involving of these orthogonal Tchebychev polynomials of second kind with respect to the Lebesgue planar measure concentrated on the ellipse.General expressions are found for the kernels polynomials associated to orthonormalized Tchebychev polynomials of second kind on the ellipse.These kernel polynomials can be used to describe the approximation of continuous functions and to solve some area extremal problems by Tchebychev polynomials of second kind on the ellipse .They can be used for the representation of the n-th partial sum of the Fourier series expansion of orthonormalized Tchebychev polynomials of second kind in the form of an integral.