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Rajkamal Nailwal

Publications and source records attributed to Rajkamal Nailwal.

9 recordsLinked to original sources
arXiv2026

Truncated Moment Problems and the Extension Property on Monomial Curves

In several papers, Stochel and Szafraniec studied moment problems on algebraic sets from an operator-theoretic perspective, investigating when positive definite sequences satisfying polynomial relations admit representing measures. Within this framework, Stochel introduced type A sets, and Bisgaard classified the plane curves defined by relations between two monomials that have this property. Curto and Fialkow introduced a stronger, truncated version of the type A property, requiring that the existence of a positive semidefinite extension of prescribed degree guarantees the existence of a representing measure. Motivated by Bisgaard's classification, we determine which plane curves defined by relations between two monomials satisfy this extension property. In the affirmative cases, we obtain explicit bounds on the required extension degree. In the negative cases, we construct truncated sequences that admit positive semidefinite extensions of arbitrarily high order but have no representing measure supported on the curve. These constructions yield explicit polynomials that are nonnegative on the corresponding curves but are not sums of squares in their coordinate rings. In the affirmative cases, we also derive explicit degree bounds for sums-of-squares certificates of strictly positive polynomials.

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arXiv2026

Cyclic polynomials in Dirichlet-type Spaces of the unit bidisk

For $α\in \mathbb{R},$ we consider the scale of function spaces, namely the Dirichlet-type space ${D}_α$ consisting of holomorphic functions on the unit bidisk $\mathbb{D}^2$, $f(z,w)=\sum_{k,l=0}^{\infty}a_{kl}z^kw^l$ such that $$\sum_{k,l=0}^{\infty}(k+l+1)^α|a_{kl}|^2 < \infty.$$ In this paper, we solve an open problem posed by Torkinejad Ziarati concerning the cyclicity of the polynomial $2-z_1-z_2$ in $ D_α$ for $ \frac32 < α\leq 2$. We provide an affirmative answer and, as a consequence, complete the characterization of cyclic polynomials in $ D_α$.

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arXiv2026

Invariant Subspaces and the $C_{00}$-Property of $3$-Brownian Shifts

In this paper, we introduce a $3$-Brownian shift $T_{σ, θ}$ on the Hilbert space $H^2(\mathbb D^2)\oplus H^2(\mathbb D)\oplus \mathbb C,$ which is a natural extension of the classical Brownian shift $B_{σ, θ}$ on $H^2(\mathbb D)\oplus \mathbb C$. This is motivated by Brownian extensions in the context of 3-isometries recently developed by A. Crăciunescu and L. Suciu. We investigate the problem of unitary equivalence for $3$-Brownian shifts on invariant subspaces of the type $M_0 \oplus M_1,$ where $ M_0 \subseteq H^2(\mathbb D^2)$ and $ M_1 \subseteq H^2(\mathbb D)\oplus \mathbb C.$ Here, $M_1$ turns out to be an invariant subspace of the respective Brownian shift $B_{σ, θ}$. We also study the asymptotic behaviour of the normalized $3$-Brownian shifts. This work is motivated by Richter \cite{R88} and very recently by work on Brownian shift on $H^2(\mathbb D)\oplus \mathbb C$ in \cite{DDS2025}.

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arXiv2025

A characterization of completely alternating functions

In this article, we characterize completely alternating functions on an abelian semigroup $S$ in terms of completely monotone functions on the product semigroup $S\times \mathbb Z_+$. We also discuss completely alternating sequences induced by a class of rational functions and obtain a set of sufficient conditions (in terms of it's zeros and poles) to determine them. As an application, we show a complete characterization of several classes of completely monotone functions on $\mathbb Z_+^2$ induced by rational functions in two variables. We also derive a set of necessary conditions for the complete monotonicity of the sequence $\{\prod_{i=1}^{k}\frac{(n+a_i)}{(n+b_i)}\}_{n \in \mathbb Z_+}, a_i, b_i \in (0,\infty)$

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arXiv2025

Rank one perturbations of 2-isometries

In this paper, we investigate when a rank-one perturbation of a $2$-isometry remains a $2$-isometry. As an application, we identify several classes of $2$-isometries on function spaces.

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arXiv2024

Gaussian Quadratures with prescribed nodes via moment theory

Let $μ$ be a positive Borel measure on the real line and let $L$ be the linear functional on univariate polynomials of bounded degree, defined as integration with respect to $μ$. In 2020, Blekherman et al., the characterization of all minimal quadrature rules of $μ$ in terms of the roots of a bivariate polynomial is given and two determinantal representations of this polynomial are established. In particular, the authors solved the question of the existence of a minimal quadrature rule with one prescribed node, leaving open the extension to more prescribed nodes. In this paper, we solve this problem using moment theory as the main tool.

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arXiv2024

The truncated univariate rational moment problem

Given a closed subset $K$ in $\mathbb{R}$, the rational $K$-truncated moment problem ($K$-RTMP) asks to characterize the existence of a positive Borel measure $μ$, supported on $K$, such that a linear functional $\mathcal{L}$, defined on all rational functions of the form $\frac{f}{q}$, where $q$ is a fixed polynomial with all real zeros of even order and $f$ is any real polynomial of degree at most $2k$, is an integration with respect to $μ$. The case of a compact set $K$ was solved by Chandler in 1994, but there is no argument that ensures that $μ$ vanishes on all real zeros of $q$. An obvious necessary condition for the solvability of the $K$-RTMP is that $\mathcal{L}$ is nonnegative on every $f$ satisfying $f|_{K}\geq 0$. If $\mathcal{L}$ is strictly positive on every $0\neq f|_{K}\geq 0$, we add the missing argument from Chandler's solution and also bound the number of atoms in a minimal representing measure. We show by an example that nonnegativity of $\mathcal{L}$ is not sufficient and add the missing conditions to the solution. We also solve the $K$-RTMP for unbounded $K$ and derive the solutions to the strong truncated Hamburger moment problem and the truncated moment problem on the unit circle as special cases.

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arXiv2023

Joint complete monotonicity of rational functions in two variables and toral $m$-isometric pairs

We discuss the problem of classifying polynomials $p : \mathbb R^2_+ \rightarrow (0, \infty)$ for which $\frac{1}{p}=\{\frac{1}{p(m, n)}\}_{m, n \geq 0}$ is joint completely monotone, where $p$ is a linear polynomial in $y.$ We show that if $p(x, y)=a+b x+c y+d xy$ with $a > 0$ and $b, c, d \geq 0,$ then $\frac{1}{p}$ is joint completely monotone if and only if $a d - b c \leq 0.$ We also present an application to the Cauchy dual subnormality problem for toral $3$-isometric weighted $2$-shifts.

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