Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0709.2935

An Extension to an Algebraic Method for Linear Time-Invariant System and Network Theory: The full AC-Calculus

Eberhard H. -A. Gerbracht
View original source ↗Read source document ↗Export metadata ↓

Abstract

Being inspired by phasor analysis in linear circuit theory, and its algebraic counterpart - the AC-(operational)-calculus for sinusoids developed by W. Marten and W. Mathis - we define a complex structure on several spaces of real-valued elementary functions. This is used to algebraize inhomogeneous linear ordinary differential equations with inhomogenities stemming from these spaces. Thus we deduce an effective method to calculate particular solutions of these ODEs in a purely algebraic way.

Explore related subjects

math.CAcs.SC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Eberhard H. -A. Gerbracht. 2007-09-19. An Extension to an Algebraic Method for Linear Time-Invariant System and Network Theory: The full AC-Calculus. https://arxiv.org/abs/0709.2935

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

KEEP EXPLORING

Related papers

arXiv2026

Reverse square function estimates for degenerate curves and their applications

Building on the classical work of Córdoba--Fefferman and the recent work of Schippa, we establish $L^4$ reverse square function estimates for functions whose Fourier support is contained in a $δ$-neighborhood of the curve $\{(ξ,|ξ|^a): |ξ|\leq 1\}$ in $\mathbb{R}^2$, for all exponents $a\in(0,\infty)\backslash\{1\}$. As applications, we derive sharp $L^4$ Strichartz estimates on the one-dimensional torus for fractional Schrödinger equations and establish new local smoothing estimates in modulation spaces. In the latter application, orthogonal Strichartz-type estimates also play a crucial role.

math.CA↗
arXiv2026

Gaussian rigidity and width of measures

Given $0\le m<n$ and a positive Radon measure $μ$ on $\mathbb R^n$ with at most polynomial growth, assume that it has finite $m$-dimensional Gaussian discrepancy, defined as $$\mathcal D(μ):=\limsup_{R\to\infty}\Bigl[R^2\sup_{x\in\mathrm{spt}μ}|G_R^μ(x)-1|\Bigr],$$ where $G_R^μ(x):=(4πR^2)^{-m/2}\int_{\mathbb R^n} e^{-|z-x|^2/(4R^2)}\,dμ(z)$. Provided that $μ$ admits a flat blow-down $\mathcal H^m\llcorner V$ (an assumption that cannot be dropped entirely), we prove that there exists a finite (smallest) $0\leδ<\infty$, the so-called width of $μ$, such that $$\mathrm{spt}μ\subseteq \bar B_δ(V+x_0),\quad δ^2\le C(m,n)\mathcal{D}(μ),$$ where $\bar B_δ(V+x_0)$ denotes the closed $δ$-neighborhood of a suitable translate of $V$. Note that the exponent is optimal because of scale invariance. In particular, when $\mathcal D(μ)=0$, the support of $μ$ is included in an $m$-dimensional affine plane. We use this fact to give a simpler proof of Preiss's rectifiability theorem: while still based on Gaussian averages of $μ$, this new proof avoids showing uniqueness (and hence conicality) of blow-downs of $m$-uniform measures, as well as studying Gaussian-weighted moments beyond second order. The strategy of proof consists of an anisotropic blow-down of $μ$, followed by an analysis of the resulting Young measure, using a rigidity lemma involving the heat semigroup.

math.CA↗
arXiv2026

Fourier decay and endpoint estimates for lacunary maximal operators

We study endpoint estimates for lacunary maximal operators associated with compactly supported measures. For every $d\geq1$, we construct a compactly supported, absolutely continuous probability measure on $\mathbb R^d$ with polynomial Fourier decay whose associated lacunary maximal operator fails to map $H^1(\mathbb R^d)$ boundedly into $L^{1,\infty}(\mathbb R^d)$. In particular, it fails to be of weak type $(1,1)$. This gives a negative answer to the weak type $(1,1)$ question posed by Seeger and Wright in 2011 and to Open Problems 14 and 15, contributed by Wright at the 2016 Conference in Harmonic Analysis, held at the University of Wisconsin--Madison in honor of Professor Michael Christ.

math.CA↗