arXiv subjects

Prashant Goyal

Publications and source records attributed to Prashant Goyal.

2 recordsLinked to original sources

arXiv2021

Random Sampling in Reproducing Kernel Subspace of Mixed Lebesgue Spaces

In this article, we consider the random sampling in the image space $V$ of mixed Lebesgue space $L^{p,q}(\mathbb{R}^{n+1})$ under an idempotent integral operator. We assume some decay and regularity conditions of the kernel and approximate the unit sphere in $V$ on a bounded cube $C_{R,S}$ by a finite-dimensional subspace of $V$. Consequently, the set of concentrated functions is totally bounded. We prove with an overwhelming probability that the random sample set uniformly distributed over $C_{R,S}$ is a stable set of sampling for the set of concentrated functions on $C_{R,S}$. Moreover, we propose an iterative scheme to reconstruct the concentrated signal from its random measurements.

math.FA↗

arXiv2018

Global Attractor For Weakly Damped, Forced Mkdv Equation Below Energy Space

We prove the existence of the global attractor in $ \dot H^s$, $s > 11/12$ for the weakly damped and forced mKdV on the one dimensional torus. The existence of global attractor below the energy space has not been known, though the global well-posedness below the energy space is established. We directly apply the I-method to the damped and forced mKdV, because the Miura transformation does not work for the mKdV with damping and forcing terms. We need to make a close investigation into the trilinear estimates involving resonant frequencies, which are different from the bilinear estimates corresponding to the KdV.

math.AP↗