Let f and g be analytic functions on the open unit disk ${\mathbb D}$ such that $|f|=|g|$ on a set A. We give an alternative proof of the result of Perez that there exists c in the unit circle ${\mathbb T}$ such that $f=cg$ when A is the union of two lines in ${\mathbb D}$ intersecting at an angle that is an irrational multiple of $\pi $, and from this, deduce a sequential generalization of the result. Similarly, the same conclusion is valid when f and g are in the Nevanlinna class and A is the union of the unit circle and an interior circle, tangential or not. We also provide sequential versions of this result and analyze the case $A=r{\mathbb T}$. Finally, we examine the most general situation when there is equality on two distinct circles in the disk, proving a result or counterexample for each possible configuration.

We discuss the concept of inner function in reproducing kernelHilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to ${1}/{f}\;$ , 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 modiûed to produce inner functions.

An analytic function f in the unit disc belongs to F(p,q,s), if

is uniformly bounded for all a ∈ . Here is the Green function of , and φa(z)=(a−z)/(1−āz). It is shown that for 0 < γ < ∞ and |w|=1 the singular inner function exp(γ(z+w)/(z−w)) belongs to F(p,q,s), 0<s≤1, if and only if . Moreover, it is proved that, if 0<s<1, then an inner function belongs to the Möbius invariant Besov-type space for some (equivalently for all) p > max{s,1−s} if and only if it is a Blaschke product whose zero sequence {zn} satisfies .

A model subspace ${{K}_{\Theta }}$ of the Hardy space ${{H}^{2}}={{H}^{2}}\left( {{\mathbb{C}}_{+}} \right)$ for the upper half plane ${{\mathbb{C}}_{+}}$ is ${{H}^{2}}\left( {{\mathbb{C}}_{+}} \right)\ominus \Theta {{H}^{2}}\left( {{\mathbb{C}}_{+}} \right)$ where $\Theta $ is an inner function in ${{\mathbb{C}}_{+}}$ . A function $\omega :\,\mathbb{R}\mapsto [0,\,\infty )$ is called an admissible majorant for ${{K}_{\Theta }}$ if there exists an $f\,\in \,{{K}_{\Theta }},\,f\,\not{\equiv }\,0,\,|f\left( x \right)|\,\le \,\omega \left( x \right)$ almost everywhere on $\mathbb{R}$ . For some (mainly meromorphic) $\Theta $ 's some parts of Adm $\Theta $ (the set of all admissible majorants for ${{K}_{\Theta }}$ ) are explicitly described. These descriptions depend on the rate of growth of arg $\Theta $ along $\mathbb{R}$ . This paper is about slowly growing arguments (slower than $x$ ). Our results exhibit the dependence of Adm $B$ on the geometry of the zeros of the Blaschke product $B$ . A complete description of Adm $B$ is obtained for $B$ 's with purely imaginary (“vertical”) zeros. We show that in this case a unique minimal admissible majorant exists.

This paper is a continuation of $[6]$ . We consider the model subspaces ${{K}_{\Theta }}={{H}^{2}}\ominus \Theta {{H}^{2}}$ of the Hardy space ${{H}^{2}}$ generated by an inner function $\Theta $ in the upper half plane. Our main object is the class of admissible majorants for ${{K}_{\Theta }}$ , denoted by Adm $\Theta $ and consisting of all functions $\omega $ defined on $\mathbb{R}$ such that there exists an $f\ne 0,f\in {{K}_{\Theta }}$ satisfying $|f\left( x \right)|\,\le \,\omega \left( x \right)$ almost everywhere on $\mathbb{R}$ . Firstly, using some simple Hilbert transform techniques, we obtain a general multiplier theorem applicable to any ${{K}_{\Theta }}$ generated by a meromorphic inner function. In contrast with $[6]$ , we consider the generating functions $\Theta $ such that the unit vector $\Theta \left( x \right)$ winds up fast as $x$ grows from $-\infty \,\text{to}\,\infty $ . In particular, we consider $\Theta \,=\,B$ where $B$ is a Blaschke product with “horizontal” zeros, i.e., almost uniformly distributed in a strip parallel to and separated from $\mathbb{R}$ . It is shown, among other things, that for any such $B$ , any even $\omega $ decreasing on $\left( 0,\,\infty\right)$ with a finite logarithmic integral is in Adm $B$ (unlike the “vertical” case treated in $[6]$ ), thus generalizing (with a new proof) a classical result related to Adm $\exp \left( i\sigma z \right),\,\sigma \,>\,0$ . Some oscillating $\omega $ 's in Adm $B$ are also described. Our theme is related to the Beurling-Malliavin multiplier theorem devoted to Adm $\exp \left( i\sigma z \right),\,\sigma \,>\,0$ , and to de Branges’ space $H\left( E \right)$ .

Let C be a separable Hilbert Space, and let Λ be the halfplane {(m, n) ∈ Ζ 2 : m ≥ 1} ∪ {(0, n) ∈ Ζ 2 : n ≥ 0} of the integer lattice. Consider the subspace ℳc (Λ) of on the torus spanned by the C-valued trigonometric functions {Ceims+int : с ∈ C, (m, n) ∈ Λ}. The notion of a Λ-analytic operator on ℳc (Λ) is defined with respect to the family of shift operators {Smn }Λ on ℳC (Λ) given by (Smnƒ)(eis , eit ) = eims+intƒ(eis , eit ). The corresponding concepts of inner function, outer function and analytic range function are explored. These ideas are applied to the spectral factorization problem in prediction theory.