[Paper Review] Majorants of meromorphic functions with fixed poles

Anton Baranov, Alexander Borichev+1 more|arXiv (Cornell University)|May 2, 2006

Holomorphic and Operator Theory10 references10 citations

TL;DR

This paper establishes sharp conditions for admissible majorants of meromorphic functions in model subspaces $ K_B $ associated with Blaschke products $ B $ having fixed poles in the lower half-plane. Using Hilbert transform techniques and quasianalyticity theorems, it proves that if the inner function's argument grows slowly (e.g., zeros are sparse or have power-type growth), then majorants decaying faster than any polynomial are not admissible unless a logarithmic integral condition is satisfied, extending the Beurling–Malliavin theory to model subspaces beyond Paley–Wiener spaces.

ABSTRACT

Let $B$ be a meromorphic Blaschke product in the upper half-plane with zeros $z_n$ and let $K_B=H^2\ominus BH^2$ be the associated model subspace of the Hardy class. In other words, $K_B$ is the space of square summable meromorphic functions with the poles at the points $\bar z_n$. A nonnegative function $w$ on the real line is said to be an admissible majorant for $K_B$ if there is a non-zero function $f\in K_B$ such that $|f|\le w$ a.e. on $\mathbb{R}$. We study the relations between the distribution of the zeros of a Blaschke product $B$ and the class of admissible majorants for the space $K_B$.

Motivation & Objective

• To characterize the class of admissible majorants for model subspaces $ K_B $ associated with meromorphic Blaschke products with fixed poles.
• To extend the Beurling–Malliavin theory on majorants in Paley–Wiener spaces to more general model subspaces $ K_B $ in the Hardy space $ H^2 $.
• To investigate how the distribution of zeros of the Blaschke product $ B $ affects the existence of nontrivial majorants for $ K_B $.
• To determine sharp decay conditions on majorants $ w $ such that $ |f| \leq w $ a.e. on $ \mathbb{R} $ for some nonzero $ f \in K_B $.

Proposed method

• Use of the Hilbert transform of the function $ \Omega = -\log w $ to analyze majorant admissibility in $ K_B $, extending techniques from Havin and Mashreghi.
• Application of the Denjoy–Carleman quasianalyticity theorem to show that if derivatives of the Fourier transform vanish at zero, the function must be identically zero.
• Construction of auxiliary functions $ g \in K_{B^\#} $ via Blaschke product perturbations to transfer admissibility conditions from $ K_B $ to $ K_{B^\#} $, where $ B^\# $ has shifted zeros.
• Use of Legendre transform arguments to relate the growth of coefficients $ \sqrt{y_n} $ to the decay of majorants via the function $ Y(r) = -\log y(r) $.
• Analysis of the Fourier transform $ F(x) = \int f(t)e^{-itx}dt $ for $ f \in K_B $, showing smoothness and decay conditions that imply $ f \equiv 0 $ if majorant decays too fast.
• Establishment of necessary and sufficient conditions via divergence of series like $ \sum n^{-3/2} \log(1/y_n) $, linking zero distribution to admissibility.

Experimental results

Research questions

• RQ1 Under what conditions on the zero sequence $ \{z_n\} $ of a Blaschke product $ B $ is a majorant $ w $ admissible for the model subspace $ K_B $?
• RQ2 How does the decay rate of a majorant $ w $ on $ \mathbb{R} $ relate to the distribution of poles $ \overline{z}_n $ in $ K_B $?
• RQ3 Can the Beurling–Malliavin theorem on majorants in Paley–Wiener spaces be extended to model subspaces $ K_B $ with non-uniformly distributed zeros?
• RQ4 What role does the convexity of $ Y(e^x) = -\log y(e^x) $ play in determining the admissibility of rapidly decaying majorants?
• RQ5 When does the vanishing of all derivatives of the Fourier transform of $ f \in K_B $ at zero imply $ f \equiv 0 $, and how is this tied to majorant decay?

Key findings

• If the sequence $ \{y_n\} $ of imaginary parts of zeros $ z_n = n + i y_n $ satisfies $ \sum n^{-3/2} \log(1/y_n) < \infty $, then any even, nonincreasing majorant $ w $ with $ \mathcal{L}(w) < \infty $ is admissible for $ K_{B_1} $.
• If $ \sum n^{-3/2} \log(1/y_n) = \infty $ and $ Y(r) = -\log y(r) $ is convex with $ Y(e^x) $ convex on $ \mathbb{R} $, then no majorant $ w $ decaying faster than any polynomial is admissible for $ K_{B_1} $.
• The function $ g(t) = \sum \sqrt{y_n} c_n / (t - n + i(y_n + 1)) $, constructed from $ f \in K_B $, satisfies $ |g(t)| \leq t^A e^{-Y(t)} $ for $ t > 1 $, linking coefficient decay to majorant decay.
• If the Fourier transform $ F(x) $ of $ f \in K_B $ satisfies $ F^{(k)}(0) = 0 $ for all $ k \geq 0 $, and $ \int_1^\infty r^{-2} \log T(r) dr = \infty $, then $ f \equiv 0 $, implying no such $ f $ exists for rapidly decaying $ w $.
• The classical Denjoy–Carleman quasianalyticity theorem is applied to show that if $ \sup_n \sqrt{y_n} (en)^k \leq \exp(\sup_r [k \log r - \frac{1}{2} Y(r/e)]) $, then $ f \equiv 0 $ under divergence of the logarithmic integral.
• The result extends to Blaschke products with zeros on the imaginary axis or with power-type growth, showing that admissibility fails for majorants decaying faster than any polynomial unless the zero distribution satisfies a logarithmic convergence condition.