This work studies optimal polynomial approximants (OPAs) in the classical Hardy spaces on the unit disk, Hp (1 < p < infinity). In particular, we uncover some estimates concerning the OPAs of degree zero and one. It is also shown that if f is an element of Hp is an inner function, or if p > 2 is an even integer, then the roots of the nontrivial OPA for 1/fare bounded from the origin by a distance depending only on p. For p not equal 2, these results are made possible by the novel use of a family of inequalities which are derived from a Banach space analogue of the Pythagorean theorem.

The well-known proof of Beurling's Theorem in the Hardy space H-2, which describes all shift-invariant subspaces, rests on calculating the orthogonal projection of the unit constant function onto the subspace in question. Extensions to other Hardy spaces H-P for 0 < p < infinity are usually obtained by reduction to the H & sup2; case via inner-outer factorization of H-P functions. In this article, we instead explicitly calculate the metric projection of the unit constant function onto a shift-Invariant subspace of the Hardy space HP when 1 < p < infinity This problem is equivalent to finding the best approximation in HP of the conjugate of an inner function. In H & sup2;, this approximation is always a constant, but in H-P, when p not equal 2, this approximation turns out to be zero or a non-constant outer function. Further, we determine the exact distance between the unit constant and any shift-invariant subspace and propose some open problems. Our results use the notion of Birkhoff-James orthogonality and Pythagorean inequalities, along with an associated dual extremal problem, which leads to some interesting inequalities. Further consequences shed light on the lattice of shift-Invariant subspaces of H-P, as well as the behavior of the zeros of optimal polynomial approximants in H-P.

This work explores several aspects of interpolating sequences for ℓ A p \ell ^p_A , the space of analytic functions on the unit disk with p p -summable Maclaurin coefficients. Much of this work is communicated through a Carlesonian lens. We investigate various analogues of Gramian matrices, for which we show boundedness conditions are necessary and sufficient for interpolation, including a characterization of universal interpolating sequences in terms of Riesz systems. We also discuss weak separation, giving a characterization of such sequences using a generalization of the pseudohyperbolic metric. Lastly, we consider Carleson measures and embeddings.

This work studies optimal polynomial approximants (OPAs) in the classical Hardy spaces on the unit disk, H^p (1 < p < ∞). For fixed f∈ H^p and n∈ℕ, the OPA of degree n associated to f is the polynomial which minimizes the quantity qf-1_p over all complex polynomials q of degree less than or equal to n. We begin with some examples which illustrate, when p≠2, how the Banach space geometry makes these problems interesting. We then weave through various results concerning limits and roots of these polynomials, including results which show that OPAs can be witnessed as solutions of certain fixed point problems. Finally, using duality arguments, we provide several bounds concerning the error incurred in the OPA approximation.

This work explores several aspects of interpolating sequences for $\ell^p_A$, the space of analytic functions on the unit disk with $p$-summable Maclaurin coefficients. Much of this work is communicated through a Carlesonian lens. We investigate various analogues of Gramian matrices, for which we show boundedness conditions are necessary and sufficient for interpolation, including a characterization of universal interpolating sequences in terms of Riesz systems. We also discuss weak separation, giving a characterization of such sequences using a generalization of the pseudohyperbolic metric. Lastly, we consider Carleson measures and embeddings.

We give a generalization of the notion of finite Blaschke products from the perspective of generalized inner functions in various reproducing kernel Hilbert spaces. Further, we study precisely how these functions relate to the so-called Shapiro–Shields functions and shift-invariant subspaces generated by polynomials. Applying our results, we show that the only entire inner functions on weighted Hardy spaces over the unit disk are multiples of monomials, extending recent work of Cobos and Seco.

This paper discusses the convexity of the range of the Berezin transform. For a bounded operator T acting on a reproducing kernel Hilbert space H (on a set X), this is the set B(T) := {< T(k)over cap(x), (k)over cap(x) >(H) : x is an element of X}, where (k)over cap(x) is the normalized reproducing kernel for H at x is an element of X. Primarily, we focus on characterizing convexity of this range for a class of composition operators acting on the Hardy space of the unit disk. (c) 2022 Elsevier Inc. All rights reserved.

For various Hilbert spaces of analytic functions on the unit disk, we characterize when a function $f$ has optimal polynomial approximants given by truncations of a single power series. We also introduce a generalized notion of optimal approximant and use this to explicitly compute orthogonal projections of 1 onto certain shift invariant subspaces.

We give a generalization of the notion of finite Blaschke products from the perspective of generalized inner functions in various reproducing kernel Hilbert spaces. Further, we study precisely how these functions relate to the so-called Shapiro–Shields functions and shift-invariant subspaces generated by polynomials. Applying our results, we show that the only entire inner functions on weighted Hardy spaces over the unit disk are multiples of monomials, extending recent work of Cobos and Seco.

Abstract For p ∈ (1, ∞) \ {2}, some properties of the space ℳ p of multipliers on ℓp A are derived. In particular, the failure of the weak parallelogram laws and the Pythagorean inequalities is demonstrated for ℳ p . It is also shown that the extremal multipliers on the ℓp A spaces are exactly the monomials, in stark contrast to the p = 2 case.

A two-point algebra is a set of bounded analytic functions on the unit disk that agree at two distinct points $a,b \in \mathbb{D}$. This algebra serves as a multiplier algebra for the family of Hardy Hilbert spaces $H^2_t := \{ f\in H^2 : f(a)=tf(b)\}$, where $t\in \mathbb{C}\cup\{\infty\}$. We show that various spectra of certain Toeplitz operators acting on these spaces are connected.

We present an approach to analyze $C^1(\mathbb{R}^m)$ functions that addresses limitations present in the Active Subspaces (AS) method of Constantine et al.(2015; 2014). Under appropriate hypotheses, our Active Manifolds (AM) method identifies a 1-D curve in the domain (the active manifold) on which nearly all values of the unknown function are attained, and which can be exploited for approximation or analysis, especially when $m$ is large (high-dimensional input space). We provide theorems justifying our AM technique and an algorithm permitting functional approximation and sensitivity analysis. Using accessible, low-dimensional functions as initial examples, we show AM reduces approximation error by an order of magnitude compared to AS, at the expense of more computation. Following this, we revisit the sensitivity analysis by Glaws et al. (2017), who apply AS to analyze a magnetohydrodynamic power generator model, and compare the performance of AM on the same data. Our analysis provides detailed information not captured by AS, exhibiting the influence of each parameter individually along an active manifold. Overall, AM represents a novel technique for analyzing functional models with benefits including: reducing $m$-dimensional analysis to a 1-D analogue, permitting more accurate regression than AS (at more computational expense), enabling more informative sensitivity analysis, and granting accessible visualizations(2-D plots) of parameter sensitivity along the AM.