Chebyshev–Blaschke products: Solutions to certain approximation problems and differential equations

In this paper, we study a special kind of finite Blaschke products called Chebyshev–Blaschke products fn,τfn,τ which can be defined by the Jacobi cosine function cd(u,τ), where τ∈R+iτ∈R+i. We will show that Chebyshev–Blaschke products solve a number of approximation problems, which are related to Zolotarev’s 3rd and 4th problems. More importantly, such a Chebyshev–Blaschke product fn,τfn,τ will be shown to be the finite Blaschke product of degree nn which has the least deviation from zero on [−k(τ),k(τ)], where k(τ)k(τ) is the elliptic modulus. Moreover, certain differential equations for Chebyshev–Blaschke products will be derived.