Chebyshev-Blaschke products: solutions to certain approximation problems and differential equations (Q464644)

In this paper, a special kind of finite Blaschke products called Chebyshev-Blaschke products is studied. Let \(k(\tau)\) be the elliptic modulus and \(\text{cd}(u,\tau)\) be the Jacobi cosine function, where \(\tau\in\mathbb{R}i\). Then the Chebyshev-Blaschke product can be defined by the following parametric equations NEWLINE\[NEWLINE f_{n,\tau}(z)=\sqrt{k(n\tau)}\,\text{cd}(nu\omega_1(n\tau),n\tau),\quad z=\sqrt{k(\tau)}\,\text{cd}(u\omega_1(\tau),\tau)\,. NEWLINE\]NEWLINE Let \(\mathcal{B}_n\) denote the set of all finite Blaschke products of degree \(n\). The authors consider the following problem. Find \(B^*_{n,\tau}\in\mathcal{B}_n\) that attains the minimum NEWLINE\[NEWLINE \sigma_{n,\tau}=\min\limits_{B_n\in\mathcal{B}_n}\max\limits_{z\in E}|B_n(z)|\,, NEWLINE\]NEWLINE where \(E=\big[-\sqrt{k(\tau)},\sqrt{k(\tau)}\big]\). It is shown that the Chebyshev-Blaschke product \(f_{n,\tau}\) is the solution to the problem with \(\sigma_{n,\tau}=\sqrt{k(n\tau)}\). Certain differential equations for Chebyshev-Blaschke products are obtained.