arXiv subjects
Publications and source records attributed to R. K. Srivastava.
We develop a structural classification of multipliers between generalized Toeplitz kernels, extending the work of Fricain and Rupam. Our results establish new equivalences between multiplier space and Carleson-type embeddings, linking them to Beurling Malliavin densities, Pólya sequences, and the spectral theory of entire functions.
Since $(\mathbb{H}^n\rtimes U(n),U(n))$ is a Gelfand pair, an exact analogue of the Heisenberg group result due to Narayanan and Ratnakumar is not possible for the Heisenberg motion group. In this article, we prove that if the Weyl transform of a finitely supported integrable function on the Heisenberg motion group is non-zero only for finitely many Fourier-Wigner pieces and have finite rank, then the function must be zero. We also prove an analogue of the Heisenberg group result on the quaternion Heisenberg group. In the end, a quantitative interpretation of these results is described through strong annihilating pair for the Weyl transform.
In this article, we study the boundedness and several properties of the quaternion Wigner transform. Using the quaternion Wigner transform as a tool, we define the quaternion Weyl transform (QWT) and prove that the QWT is compact for a certain class of symbols in $L^{r}\left(\mathbb{R}^{4}, \mathbb{Q}\right)$ with $1 \leq r \leq 2.$ Moreover, it can not be extended as a bounded operator for symbols in $L^{r}\left(\mathbb{R}^{4},\mathbb{Q}\right)$ for $2<r<\infty.$ In addition, we prove a rank analogue of the Benedicks-Amrein-Berthier theorem for the QWT. Further, we remark about the set of injectivity and Helgason's support theorem for the quaternion twisted spherical means.
In this article, we study the injectivity of the spherical mean for continuous functions on the Métivier group. The spherical mean is injective for $f(z, .)\in L^p(\mathbb{R}^m),~1\leq p \leq 2$ with tempered growth in $z$ variable. This result is also true for a class of functions in $L^p(\mathbb{C}^n),\,1\leq p\leq\infty$ without tempered growth. Further, we obtain a two-radii theorem for functions on the Métivier group, which are tempered in $z$ variable and periodic in the centre variable.
Let $Z_{r, R}$ be the space of continuous functions on the annulus $B_{r, R}$ in $\mathbb C^n$ whose $λ$-twisted spherical mean, in the set up of the Métivier group, vanishes over the spheres $S_s(z)\subset B_{r, R} $ with ball $B_r(0)\subseteq B_s(z).$ We characterize the spherical harmonic coefficients of functions in $Z_{r, R},$ eventually, in terms of polynomial growth, by which we infer support theorem. Further, we prove that non-harmonic complex cone and the boundary of a bounded domain are sets of injectivity for the $λ$-twisted spherical means.
In this article, we define Weyl transform on second countable type - $I$ locally compact group $G,$ and as an operator on $L^2(G),$ we prove that the Weyl transform is compact when the symbol lies in $L^p(G\times \hat{G})$ with $1\leq p\leq 2.$ Further, for the Euclidean motion group and Heisenberg motion group, we prove that the Weyl transform can not be extended as a bounded operator for the symbol belongs to $L^p(G\times \hat{G})$ with $2<p<\infty.$ To carry out this, we construct positive, square integrable and compactly supported function, on the respective groups, such that $L^{p'}$ norm of its Fourier transform is infinite, where $p'$ is the conjugate index of $p.$
In this article, we prove that (unit sphere, non-harmonic cone) is a Heisenberg uniqueness pair for the symplectic Fourier transform on $\mathbb C^n.$ We derive that spheres as well as non-harmonic cones are determining sets for the spectral projections of the finite measure supported on the unit sphere. Further, we prove that if the Fourier transform of a finitely supported function on step two nilpotent Lie group is of arbitrary finite rank, then the function must be zero. The latter result correlates to the annihilating pair for the Weyl transform.
In this article, we prove that if the group Fourier transform of certain integrable functions on the Heisenberg motion group (or step two nilpotent Lie groups) is of finite rank, then the function is identically zero. These results can be thought as an analogue to the Benedicks theorem that dealt with the uniqueness of the Fourier transform of integrable functions on the Euclidean spaces.
In this article, we prove that a cone is a Heisenberg uniqueness pair corresponding to sphere as long as the cone does not completely recline on the level surface of any homogeneous harmonic polynomial on $\mathbb R^n.$ We derive that $\left(S^2, \text{ paraboloid}\right)$ and $\left(S^2, \text{ geodesic of } S_r(o)\right)$ are Heisenberg uniqueness pairs for a class of certain symmetric finite Borel measures in $\mathbb R^3.$ Further, we correlate the problem of Heisenberg uniqueness pairs to the sets of injectivity for the spherical mean operator.
In this article, we prove that if the Fourier transform of a certain integrable function on the Euclidean motion group is of finite rank, then the function has to vanish identically. Further, we explore a new variance of the uncertainty principle, the Heisenberg uniqueness pairs on the Euclidean motion group as well as on the product group $\mathbb R^n\times K,$ where $K$ is a compact group.
Let $X(Γ)$ be the space of all finite Borel measure $μ$ in $\mathbb R^2$ which is supported on the curve $Γ$ and absolutely continuous with respect to the arc length of $Γ$. For $Λ\subset\mathbb R^2,$ the pair $\left(Γ, Λ\right)$ is called a Heisenberg uniqueness pair for $X(Γ)$ if any $μ\in X(Γ)$ satisfies $\hatμ\vert_Λ=0,$ implies $μ=0.$ We explore the Heisenberg uniqueness pairs corresponding to the cross, exponential curves, and surfaces. Then, we prove a characterization of the Heisenberg uniqueness pairs corresponding to finitely many parallel lines. We observe that the size of the determining sets $Λ$ for $X(Γ)$ depends on the number of lines and their irregular distribution that further relates to a phenomenon of interlacing of certain trigonometric polynomials.
A Heisenberg uniqueness pair is a pair $\left(Γ, Λ\right)$, where $Γ$ is a curve and $Λ$ is a set in $\mathbb R^2$ such that whenever a finite Borel measure $μ$ having support on $Γ$ which is absolutely continuous with respect to the arc length on $Γ$ satisfies $\hatμ\vert_Λ=0,$ then it is identically $0.$ In this article, we investigate the Heisenberg uniqueness pairs corresponding to the spiral, hyperbola, circle and certain exponential curves. Further, we work out a characterization of the Heisenberg uniqueness pairs corresponding to four parallel lines. In the latter case, we observe a phenomenon of interlacing of three trigonometric polynomials.
In this article, we review the Weyl correspondence of bigraded spherical harmonics and use it to extend the Hecke-Bochner identities for the spectral projections $f\timesφ_k^{n-1}$ for function $f\in L^p(\mathbb C^n)$ with $1\leq p\leq\infty.$ We prove that spheres are sets of injectivity for the twisted spherical means with real analytic weight. Then, we derive a real analytic expansion for the spectral projections $f\timesφ_k^{n-1}$ for function $f\in L^2(\mathbb C^n).$
In this article, we prove that a complex cone is a set of injectivity for the twisted spherical means for the class of all continuous functions on $\mathbb C^n$ as long as it does not completely lay on the level surface of any bi-graded homogeneous harmonic polynomial on $\mathbb C^n.$ Further, we produce examples of such level surfaces.
It is well known that a line in $\mathbb R^2$ is not a set of injectivity for the spherical means for odd functions about that line. We prove that any line passing through the origin is a set of injectivity for the twisted spherical means (TSM) for functions $f\in L^2(\mathbb C),$ whose each of spectral projection $ e^{\frac{1}{4}|z|^2}f\timesφ_k$ is a polynomial. Then, we prove that any Coxeter system of even number of lines is a set of injectivity for the TSM for $L^q(\mathbb C),~1\leq q\leq2.$
In this article, we prove that any pair of perpendicular planes is a set of injectivity for the twisted spherical means (TSM) for $L^p(\mathbb C^n) (n\geq2)$ with $1\leq p\leq2.$ Then, we imitate that any Coxeter system of even number of planes is a set of injectivity for the TSM for $L^p(\mathbb C^n).$ We further observe that a set $S_R^{2n-1}\times\mathbb C$ is a set of injectivity for the TSM for a ceratin class of functions on $\mathbb C^{n+1}.$
In this article, we show that the spheres $S_R(o)=\{z\in\mathbb C^n: |z|=R\}$ are sets of injectivity for the weighted twisted spherical means (WTSM) for a suitable class of functions on $\mathbb C^n$. The weights here are spherical harmonics on $S^{2n-1}.$ In general, the question of set of injectivity for the twisted spherical means (TSM) with real analytic weight is still open. We would like to refer to \cite{NRR}, for some results on the sets of injectivity for the spherical means with real analytic weights in the Euclidean setup.
Let $Z_{r,R}$ be the class of all continuous functions $f$ on the annulus $\Ann(r,R)$ in the real hyperbolic space $\mathbb B^n$ with spherical means $M_sf(x)=0$, whenever $s>0$ and $x\in \mathbb B^n$ are such that the sphere $S_s(x)\subset \Ann(r, R) $ and $B_r(o)\subseteq B_s(x).$ In this article, we give a characterization for functions in $Z_{r,R}$. In the case $R=\infty$, this result gives a new proof of Helgason's support theorem for spherical means in the real hyperbolic spaces.