arXiv subjects

Waleed Noor

Publications and source records attributed to Waleed Noor.

4 recordsLinked to original sources

arXiv2026

A Hardy space approximation supporting zero-free half-planes for the $\zeta$-function

An equivalent version of the B\'aez-Duarte criterion \cite{baez} for the Riemann Hypothesis (RH) by Bagchi states that the RH holds true if and only if the function $E(s) = 1/s$ belongs to the closed linear span of $G_k(s) = (k^{-s} - k^{-1})\zeta(s)/s,\, k \geq 2$ in the Hardy space $ H^2(\mathbb{C}_{1/2}) $, where $\mathbb{C}_{\alpha}$ denotes the half-plane $\mathrm{Re}(s)>\alpha$. We first show that if $E$ belongs to the closure of span$(G_k)_{k\geq 2}$ in $ H^2(\mathbb{C}_{\alpha}) $ for $\alpha>1/2$, then $\zeta$ is zero-free in $\mathbb{C}_\alpha$. We then use this as the basis for a numerical analysis of the sequence \[ s_n = \left\| \sum_{k=2}^{n} \mu(k) G_k - E \right\|^2_\alpha, \] for $1/2\leq \alpha \leq 1$, where $\left\|.\right\|_\alpha$ is the norm in $H^2(\mathbb{C}_{\alpha})$ and $\mu$ the M\"obius function.

math.FA↗

arXiv2025

Cyclicity of composition operators on the Paley-Wiener spaces

In this article we characterize the cyclicity of bounded composition operators $C_ϕf=f\circ ϕ$ on the Paley-Wiener spaces of entire functions $B^2_σ$ for $σ>0$. We show that $C_ϕ$ is cyclic precisely when $ϕ(z)=z+b$ where either $b\in\mathbb{C}\setminus\mathbb{R}$ or $b\in\mathbb{R}$ with $0<|b|\leq π/σ$. We also describe when the reproducing kernels of $B^2_σ$ are cyclic vectors for $C_ϕ$ and see that this is related to a question of completeness of exponential sequences in $L^2[-σ,σ]$. The interplay between cyclicity and complex symmetry plays a key role in this work.

math.FA↗

arXiv2024

Orthogonality questions in the Hardy space related to $ζ$-zeros

A Hardy space approach to the Nyman-Beurling and Báez-Duarte criterion for the Riemann Hypothesis (RH) was introduced recently in [18] and further developed in [13]. It states that the RH holds if and only if a particular sequence of functions $(h_k)_{k\geq 2}$ is complete in the Hardy space $H^2$. This article is concerned with orthogonality questions related to the family $(h_k)_{k\geq 2}$. The first goal is to analyze the orthogonal complement of $\mathcal{N}=\mathrm{span}(h_k)_{k\geq 2}$ in $H^2$. Unbounded Toeplitz operators on $H^p$ spaces and de Branges-Rovnyak spaces play a central role and our results show that the size and dimension of $\mathcal{N}^\perp$ reveal information on the zeros of the Riemann $ζ$-function. The second goal is to show that $(h_k)_{k\geq 2}$ possesses a complete biorthogonal sequence in $H^2$. We also discuss a folklore conjecture about the number of $ζ$-zeros if the RH fails.

math.FA↗

arXiv2022

A weighted composition semigroup related to three open problems

The semi-group of weighted composition operators $(W_n)_{n\geq 1}$ where \[ W_nf(z)=(1+z+\ldots+z^{n-1})f(z^n) \] on the classical Hardy-Hilbert space $H^2$ of the open unit disk is related to the Riemann Hypothesis (RH) (see \cite{Waleed}). The semigroup $(W_n)_{n\geq 1}$ is also closely related to the Invariant Subspace Problem (ISP) and the Periodic Dilation Completeness Problem (PDCP). We obtain results on cyclic vectors, spectra, invariant and reducing subspaces. In particular, we show that several basic questions related to the semigroup $(W_n)_{n\geq 1}$ are equivalent to the RH and provide generalizations of the Báez-Duarte criterion for the RH (see \cite{Baez-Duarte}).

math.FA↗