Abstract. We analyze the singularities of rational inner functions on the unit bidisk and study both when these functions belong to Dirichlet-type spaces and when their partial derivatives belong to Hardy spaces. We characterize derivative H p membership purely in terms of contact order, a measure of the rate at which the zero set of a rational inner function approaches the distinguished boundary of the bidisk. We also show that derivatives of rational inner functions with singularities fail to be in H p for p ≥ 3 2 and that higher nontangential regularity of a rational inner function paradoxically reduces the H p integrability of its derivative. We derive inclusion results for Dirichlet-type spaces from derivative inclusion for H p . Using Agler decompositions and local Dirichlet integrals, we further prove that a restricted class of rational inner functions fails to belong to the unweighted Dirichlet space.
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 analyze the behavior of rational inner functions on the unit bidisk near singularities on the distinguished boundary T 2 using level sets. We show that the unimodular level sets of a rational inner function can be parametrized with analytic curves and connect the behavior of these analytic curves to that of the zero set of. We apply these results to obtain a detailed description of the fine numerical stability of : for instance, we show that @ @z 1 and @ @z 2 always possess the same L p-integrability on T 2 , and we obtain combinatorial relations between intersection multiplicities at singularities and vanishing orders for branches of level sets. We also present several new methods of constructing rational inner functions that allow us to prescribe properties of their zero sets, unimodular level sets, and singularities.
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.
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