Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0705.3365

On Some Properties of Linear Mapping Induced by Linear Descriptor Differential Equation

Serhiy Zhuk
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we introduce linear mapping D from WnF\subset Ln into Lm\times Rm, induced by linear differential equation d/dt Fx(t)-C(t)x(t)=f(t),Fx(t_0)=f_0. We prove that D is closed dense defined mapping for any m\times n-matrix F. Also adjoint mapping D* is constructed and its domain WmF is described. Some kind of so-called "integration by parts" formula for vectors from WnF, WmF is suggested. We obtain a necessary and sufficient condition for existence of generalized solution of equation Dx=(f,f_0). Also we find a sufficient criterion for closureness of the R(D) in Lm\times Rm which is formulated in terms of transparent conditions for blocks of matrix C(t). Some examples are supplied to illustrate obtained results.

Explore related subjects

math.CAmath.OC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Serhiy Zhuk. 2007-05-23. On Some Properties of Linear Mapping Induced by Linear Descriptor Differential Equation. https://arxiv.org/abs/0705.3365

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

KEEP EXPLORING

Related papers

arXiv2026

A variational principle for Gaussian lattice sums

We consider a two-dimensional analog of Jacobi theta functions and prove that, among all lattices $Λ\subset \mathbb{R}^2$ with fixed density, the minimal value is maximized by the hexagonal lattice. This result can be seen as the inhomogeneous counterpart to Montgomery's 1988 result, which proved that the hexagonal lattice minimizes the maximum.

math.CA↗
arXiv2026

Dimension-free estimates for discrete maximal functions related to normalized sampled gaussians

In this paper, we investigate dimension-free estimates for convolution operators with discrete normalized sampled Gaussians (related to the Theta function) in the context of maximal, jump, $r$-variational and oscillation inequalities on $\ell^p(\mathbb{Z}^d)$ spaces. This is the first instance of a discrete maximal function in the literature - a non-semigroup example - where dimension-free $\ell^p(\mathbb{Z}^d)$ bounds are provided for the entire range of $1 < p < \infty$. The methods of proof rely on developing robust Fourier techniques, which are combined with the fractional derivative, a tool that has not been previously applied to studying similar questions in the discrete setting.

math.CA↗
arXiv2026

Characterizations of fractional operators via integral transforms

In 1974, J. S. Lew established a reasonable conjecture regarding an axiomatic characterization for the one-dimensional Riemann--Liouville integral. This conjecture was proved by Cartwright and McMullen in 1978. After that, little further work has been done on this topic, except some extensions for the Stieltjes case in one and several variables. In this paper, we prove the necessity of the axioms established in the conjecture of J. S. Lew using the Cauchy functional equation and Hamel bases. In addition, we give a proof for the characterization in several variables by employing the Titchmarsh theorem, as a natural extension of the approach of Cartwright and McMullen. We also provide an alternative version and proof in one and several variables with Laplace transforms and the Cauchy functional equation, weakening parts of the continuity assumption. We show a similar result for the Riesz potential in terms of the Fourier transform. Finally, we illustrate how the theory can be used for characterization in the context of fractional calculus with respect to a non-smooth integrator, based on transmutation and measures.

math.CA↗