Related papers: The Hilbert space basis and Hilbert's eighth probl…

Montgomery's pair correlation conjecture predicts the asymptotic behavior of the function $N(T,\beta)$ defined to be the number of pairs $\gamma$ and $\gamma'$ of ordinates of nontrivial zeros of the Riemann zeta-function satisfying…

We consider Hilbert spaces of analytic functions in the disk with a normalized reproducing kernel and such that the backward shift $f(z) \mapsto \frac{f(z)-f(0)}{z}$ is a contraction on the space. We present a model for this operator and…

Given a finite set $\sigma$ of the unit disc $\mathbb{D}$ and a holomorphic function $f$ in $\mathbb{D}$ which belongs to a class $X$ we are looking for a function $g$ in another class $Y$ which minimizes the norm $|g|_{Y}$ among all…

We study for the first time the action of the Hilbert matrix $$\mathcal H=(c_{n,k})_{n,k\geq 0}, \quad c_{n,k}=\frac{1}{n+k+1}$$ on the analytic tent spaces $AT^q_p, 1<p,q <\infty,$ of the unit disc $\mathbb D$ of the complex plane. They…

We compare and contrast results of E. Davis, of A. Bigatti, A.V. Geramita and the author, and of J. Ahn and the author. The underlying idea is that certain numerical conditions on the Hilbert function of a finite set of points in projective…

In this short note we present a far generalization of the following very well-known assertion: assume that we have two orthonormal sequences in a Hilbert space and these sequences are quadratically close to each other. Then if one of these…

In the present paper, we characterize all possible Hilbert functions of graded ideals in a polynomial ring whose regularity is smaller than or equal to $d$, where $d$ is a positive integer. In addition, we prove the following result which…

Conjugations commuting with $\mathbf{M}_z$ and intertwining $\mathbf{M}_z$ and $\mathbf{M}_{\bar z}$ in $L^2(\mathcal{H})$, where $\mathcal{H}$ is a Hilbert space, are characterized. We also investigate which of them leave invariant the…

Consider the discrete cubic Hilbert transform defined on finitely supported functions $f$ on $\mathbb{Z}$ by \begin{eqnarray*} H_3f(n) = \sum_{m \not = 0} \frac{f(n- m^3)}{m}. \end{eqnarray*} We prove that there exists $r <2$ and universal…

Let $\mathcal{H}$ be a right quaternionic Hilbert space and let $T$ be a quaternionic normal operator with the domain $\mathcal{D}(T) \subset \mathcal{H}$. Then for a fixed unit imaginary quaternion $m$, there exists a Hilbert basis…

We compute the exact value of the essential norm of a generalized Hilbert matrix operator acting on weighted Bergman spaces $A^p_v$ and weighted Banach spaces $H^\infty_v$ of analytic functions, where $v$ is a general radial weight. In…

In this short article we show an orthogonal decomposition of a Hilbert space as a sum of null solutions of the first derivative and the first derivative of a traceless higher order Hilbert/Sobolev space. We define orthogonal projections and…

We construct in a Sonine Space of entire functions a subspace related to the Riemann zeta function and we show that the quotient contains vectors intrinsically attached to the non-trivial zeros and their multiplicities.

In this note, we prove the existence of infinitely many zeros of certain generalized Hurwitz zeta functions in the domain of absolute convergence. This is a generalization of a classical problem of Davenport, Heilbronn and Cassels about the…

It is well known that the Hilbert matrix operator $\mathcal {H}$ is bounded from $H^{\infty}$ to the mean Lipschitz spaces $\Lambda^{p}_{\frac{1}{p}}$ for all $1<p<\infty$. In this paper, we prove that the range of Hilbert matrix operator…

Suppose $\alpha,\beta \in \mathbb{R}\backslash \mathbb{Z}^-$ such that $\alpha+\beta>-1$ and $1\leq p \leq \infty$. Let $u=P_{\alpha,\beta}[f]$ be an $(\alpha,\beta)$-harmonic mapping on $\mathbb{D}$, the unit disc of $\mathbb{C}$, with the…

A new Hilbert-type integral inequality in the whole plane with the non-homogeneous kernel and parameters is given. The constant factor related to the hypergeometric function and the beta function is proved to be the best possible. As…

For any Ritt operator $T$ acting on a noncommutative $L^p$-space, we define the notion of \textit{completely} bounded functional calculus $H^\infty(B_\gamma)$ where $B_\gamma$ is a Stolz domain. Moreover, we introduce the `column square…

We study $H^p$ spaces of Dirichlet series, called $\mathcal{H}^p$, for the range $0<p< \infty$. We begin by showing that two natural ways to define $\mathcal{H}^p$ coincide. We then proceed to study some linear space properties of…

Let X be a closed subscheme and let HF(X,-) and hp(X,-) denote, respectively, the Hilbert function and the Hilbert polynomial of X. We say that X has bipolynomial Hilbert function if HF(X,d)=min{hp(P^n,d),hp(X,d)} for every non-negative…