For functions $f$ in Dirichlet-type spaces we study how to determine constructively optimal polynomials $p_n$ that minimize $\|p f-1\|_\alpha$ among all polynomials $p$ of degree at most $n$. Then we give upper and lower bounds for the rate of decay of $\|p_{n}f-1\|_{\alpha}$ as $n$ approaches $\infty$. Further, we study a generalization of a weak version of the Brown-Shields conjecture and some computational phenomena about the zeros of optimal polynomials.Comment: 26 pages, 2 figures, submitted for publicatio

We study connections between orthogonal polynomials, reproducing kernel functions, and polynomials p minimizing Dirichlet-type norms pf − 1 α for a given function f . For α ∈ [0, 1] (which includes the Hardy and Dirichlet spaces of the disk) and general f , we show that such extremal polynomials are non-vanishing in the closed unit disk. For negative α, the weighted Bergman space case, the extremal polynomials are non-vanishing on a disk of strictly smaller radius, and zeros can move inside the unit disk. We also explain how distD α (1, f · Pn), where Pn is the space of polynomials of degree at most n, can be expressed in terms of quantities associated with orthogonal polynomials and kernels, and we discuss methods for computing the quantities in question.

Abstract. We discuss the concept of inner function in reproducing kernel Hilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to f , where f is a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modi ed to produce inner functions.

Abstract. We introduce a large family of reproducing kernel Hilbert spaces H ⊂ Hol(D), which include the classical Dirichlettype spaces D α , by requiring normalized monomials to form a Riesz basis for H. Then, after precisely evaluating the n-th optimal norm and the n-th approximant of f (z) = 1−z, we completely characterize the cyclicity of functions in Hol(D) with respect to the forward shift.