arXiv subjects

Erling A. T. Svela

Publications and source records attributed to Erling A. T. Svela.

5 recordsLinked to original sources

arXiv2026

A quantum harmonic analysis approach to nonlinear time-frequency concentration

We study nonlinear concentration problems for time-frequency distributions in the Cohen class. Using recent techniques from quantum harmonic analysis (QHA) we provide both positive and negative results, such as sufficient conditions for the existence of optimizers in terms of the ``window operator'' and explicit examples where the supremum is never attained. We also study the structural properties of window operators, in particular operators that yield weakly continuous concentration functionals and operators for which the nonlinear concentration problem admits an optimizer, also beyond the Heisenberg representation. We then consider generalizations to the study of concentration problems for phase space representations of operators. We consider generalized Husimi distributions via quantum convolution, and their optimization problem when optimizing over Hilbert--Schmidt and density operators. Lastly, we consider representations of operators on double phase space, in the spirit of quantum time-frequency analysis, and give a full solution in terms of the Weyl symbols.

math.FA↗

arXiv2026

On the existence of optimizers for nonlinear time-frequency concentration problems: the Born--Jordan distribution

We study the $L^p$ concentration problem for the Born--Jordan distribution in dimension $d>1$, thus extending the one-dimensional analysis in [Stra-Svela-Trapasso, J. Math. Pures Appl. (2026)]. We show that the existence of concentration optimizers depends on the exponent $p$ with a critical threshold at $p_*(d)= \frac{2d}{d-2}$ for $d\geq2$ (with the understanding that $p_*(2)=\infty$). In particular, for subcritical exponents $1\leq p p_*(d)$ we show that the functional is unbounded. We also provide the complete solution in the (significantly more) challenging critical regime in dimension $d=2$.

math.CA↗

arXiv2026

On the existence of optimizers for nonlinear time-frequency concentration problems: the Wigner distribution

We prove that, for any measurable phase space subset $Ω\subset\mathbb{R}^{2d}$ with $0<|Ω|<\infty$ and any $1\le p < \infty$, the nonlinear concentration problem $$ \sup_{f \in L^2(\mathbb{R}^d)\setminus\{0\}}\frac{\|Wf\|_{L^p(Ω)}}{\|f\|_{L^2}^2}$$ admits an optimizer, where $Wf$ is the Wigner distribution of $f$. The main obstruction is that $Wf$ is covariant (not invariant) under time-frequency shifts, which impedes weak upper semicontinuity, so the effects of constructive interference must be taken into account. We close this compactness gap via concentration compactness for Heisenberg-type dislocations, together with a new asymptotic formula that quantifies the limiting contribution to concentration over $Ω$ from asymptotically separated wave packets. When $p=\infty$ we also identify the sharp constant $2^d$ and show that it is attained. We also discuss some related extensions: For $τ$-Wigner distributions with $τ\in (0,1)$ we isolate a chain phenomenon that obstructs the same strategy beyond the Wigner case ($τ=1/2$), while for the Born-Jordan distribution in $d=1$ we obtain weak continuity, and thus existence of concentration optimizers for all $1\le p<\infty$ (the $p=\infty$ supremum equals $π$ but is not attained).

math.CA↗

arXiv2026

Extensions of Daubechies' theorem: Reinhardt domains, Hagedorn wavepackets and mixed-state localization operators

Daubechies-type theorems for localization operators are established in the multi-variate setting, where Hagedorn wavepackets are identified as the proper substitute of the Hermite functions. The class of Reinhardt domains is shown to be the natural class of masks that allow for a Daubechies-type result. Daubechies' classical theorem is a consequence of double orthogonality results for the short-time Fourier transform. We extend double orthogonality to the quantum setting and use it to establish Daubechies-type theorems for mixed-state localization operators, a key notion of quantum harmonic analysis. Lastly, we connect the results to Toeplitz operators on quantum Gabor spaces.

math.FA↗

arXiv2025

On quantum large sieve inequalities and operator recovery from incomplete information

We obtain large sieve type inequalities for the Rayleigh quotient of the restriction of phase space representations of higher rank operators, via an operator analogue of the short-time Fourier transform (STFT). The resulting bounds are referred to as `quantum large sieve inequalities'. On the shoulders of Donoho and Stark, we demonstrate that these inequalities guarantee the recovery of an operator whose phase-space information is missing or unobservable over a 'measure-sparse' region $Ω$, by solving an $L^{1}$-minimization program. This is an operator version of what is commonly known as `Logan's phenomenon'. Moreover, our results can be viewed as a deterministic, continuous variable version, on phase space, of `low-rank' matrix recovery, which itself can be regarded as an operator version of (finite-rank) compressive sensing. Our results depend on an abstract large sieve principle for operators with integrable STFT and on a non-commutative analogue of the local reproducing formula in rotationally invariant domains (first stated by Seip for the Fock space of entire functions). As an application, we obtain concentration estimates for Cohen's class distributions and the Husimi function. We motivate the paper by comparing with Nicola and Tilli's Faber-Krahn inequality for the STFT, illustrating that norm bounds on a domain $Ω$, obtained by large sieve methods, introduce a trade-off between sparsity and concentration properties of $Ω$: If $Ω$ is sparse, the large sieve bound may significantly improve known operator norm bounds, while if $Ω$ is concentrated, it produces worse bounds.

math.FA↗