arXiv subjects

Helge Knutsen

Publications and source records attributed to Helge Knutsen.

4 recordsLinked to original sources

arXiv2022

Notes on Hardy's Uncertainty Principle for the Wigner distribution and Schrödinger evolutions

We consider Schrödinger equations with real quadratic Hamiltonians, for which the Wigner distribution of the solution at a given time equals, up to a linear coordinate transformation, the Wigner distribution of the initial condition. Based on Hardy's uncertainty principle for the joint time-frequency representation, we prove a uniqueness result for such Schrödinger equations, where the solution cannot have strong decay at two distinct times. This approach reproduces known, sharp results for the free Schrödinger equation and the harmonic oscillator, and we also present an explicit scheme for quadratic systems based on positive definite matrices.

math.AP↗

arXiv2022

A Fractal Uncertainty Principle for the Short-Time Fourier Transform and Gabor multipliers

We study the fractal uncertainty principle in the joint time-frequency representation, and we prove a version for the Short-Time Fourier transform with Gaussian window on the modulation spaces. This can equivalently be formulated in terms of projection operators on the Bargmann-Fock spaces of entire functions. Specifically for signals in $L^2(\mathbb{R}^d)$, we obtain norm estimates of Daubechies' time-frequency localization operator localizing on porous sets. The proof is based on the maximal Nyquist density of such sets, and for multidimensional Cantor iterates we derive explicit upper bound asymptotes. Finally, we translate the fractal uncertainty principle to discrete Gaussian Gabor multipliers.

math.FA↗

arXiv2021

Daubechies' Time-Frequency Localization Operator on Cantor Type Sets II

We study a version of the fractal uncertainty principle in the joint time-frequency representation. Namely, we consider Daubechies' localization operator projecting onto spherically symmetric $n$-iterate Cantor sets with an arbitrary base $M>1$ and alphabet $\mathscr{A}$. We derive an upper bound asymptote up to a multiplicative constant for the operator norm in terms of the base $M$ and alphabet size $|\mathscr{A}|$ of the Cantor set. For any fixed base and alphabet size, we show that there are Cantor sets such that the asymptote is optimal. In particular, the asymptote is precise for the mid-third Cantor set, which was studied in part I. Nonetheless, this does not extend to every Cantor set as we provide examples where the optimal asymptote is not achieved.

math.FA↗

arXiv2019

Daubechies' Time-Frequency Localization Operator on Cantor Type Sets

We study Daubechies' time-frequency localization operator, which is characterized by a window and weight function. We consider a Gaussian window and a spherically symmetric weight as this choice yields explicit formulas for the eigenvalues, with the Hermite functions as the associated eigenfunctions. Inspired by the fractal uncertainty principle in the separate time-frequency representation, we define the $n$-iterate spherically symmetric Cantor set in the joint representation. For the $n$-iterate Cantor set, precise asymptotic estimates for the operator norm are then derived up to a multiplicative constant.

math.FA↗