Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2403.06169

Analyticity, superoscillations and supershifts in several variables

F. Colombo·I. Sabadini·D. C. Struppa·A. Yger
View original source ↗Read source document ↗Export metadata ↓

Abstract

Superoscillations have roots in various scientific disciplines, including optics, signal processing, radar theory, and quantum mechanics. This intriguing mathematical phenomenon permits specific functions to oscillate at a rate surpassing their highest Fourier component. A more encompassing concept, supershifts, extends the idea of superoscillations to functions that are not sum of exponential functions. This broader notion is linked to Bernstein and Lagrange approximation of analytic functions in $\mathbb{C}^n$. Recent advancements in the theory of superoscillations and supershifts in one variable have focused on their time evolution. This paper takes a step further by expanding the notion of supershifts to include the case of several variables. We provide specific examples related to harmonic analysis where the variables vary in multi-dimensional frequency (space, or scale) domains.

Explore related subjects

math.CV

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

F. Colombo, I. Sabadini, D. C. Struppa, A. Yger. 2024-03-10. Analyticity, superoscillations and supershifts in several variables. https://arxiv.org/abs/2403.06169

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Convolution Regularization Preserves the $L^2$-Estimate Property for $(1,n)$-Forms

In this paper, we prove that the \(L^2\)-estimate property for \((1,n)\)-forms is preserved under the standard convolution regularization. As applications, we show that any singular Hermitian metric satisfying the optimal or multiple coarse \(L^2\)-estimate property for \((n,1)\) or \((1,n)\)-forms is Griffiths semi-positive. This resolves a question posed by Deng--Ning--Wang and a question by Inayama.

math.CV↗
arXiv2026

Sharp Bounds for Higher-Order Schippers Functionals Associated with Lune and Bean Domains

We obtain sharp bounds for the third- and fourth-order Schippers functionals, $|σ_3(f)(0)|$ and $|σ_4(f)(0)|$, for subclasses of univalent functions associated with non-classical geometric domains. In particular, we investigate the lune-starlike class $\mathcal{S}_{\leftmoon}^*$ and the lune-convex class $\mathcal{C}_{\leftmoon}$ determined by the subordination \[ \frac{zf'(z)}{f(z)} \prec z+\sqrt{1+z^2}, \qquad 1+\frac{zf''(z)}{f'(z)} \prec z+\sqrt{1+z^2}, \] respectively, together with the bean-domain class $\mathcal{BT}_{\mathfrak{B}}$ associated with \[ \mathfrak{B}(z)=\sqrt{1+\tanh z}. \] Using Carathéodory coefficient parametrizations and extremal optimization techniques, we derive exact estimates for the higher-order Schwarzian derivatives at the origin and identify the corresponding extremal functions. In addition, geometric descriptions of the associated extremal image domains are provided to illustrate the sharpness phenomena. The obtained results further yield sharp bounds for the initial Grunsky coefficients $g_{1,1}$ and $g_{1,2}$. These findings provide a precise description of higher-order Schwarzian structures for univalent functions related to lune and bean shaped domains.

math.CV↗
arXiv2026

Cesàro operator induced by a Bergman kernel

Let $μ$ be a positive Borel measure on $[0,1)$ and $ω$ a radial weight. In this paper we consider the Cesàro-type operator $C_{μ,ω}$ induced by the reproducing kernel $B^ω$ of the weighted Bergman space $A^2_ω$, given by $$ C_{μ,ω}(f)(z)=\int_{0}^{1}f(tz)B^ω_t(z)\,dμ(t), \quad z \in \mathbb{D}, $$ for functions $f$ analytic in $\mathbb{D}$. Under the assumption that $ω$ satisfies a natural doubling property, we study the boundedness of $C_{μ,ω}$ acting on several spaces of analytic functions, including Hardy spaces $H^p$ and weighted Bergman spaces $A^p_ν$. For $0<p,q<\infty$ and a two-sided doubling weight $ν$, we completely characterize when $C_{μ,ω}: H^p \to H^q$ and $C_{μ,ω}: A^p_ν \to A^q_ν$ are bounded in terms of the interplay of tail integrals or moments of the inducing weights and the measure $μ$. Many of the results obtained are new even in the setting of standard weights or when the Bergman reproducing kernel is replaced by the Cauchy kernel. In addition, we consider $C_{μ,ω}$ acting on $H^{\infty}$, Korenblum spaces and weighted Hardy spaces.

math.CV↗