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It is an established fact that a finite difference operator approximates a derivative with a fixed algebraic rate of convergence. Nevertheless, we exhibit a new finite difference operator and prove it has spectral accuracy. Its rate of…
The inverse problem for the Sturm- Liouville operator with complex periodic potential and discontinuous coefficients on the axis is studied. Main characteristics of the fundamental solutions are investigated, the spectrum of the operator is…
In this paper, we develop two approaches to investigation of inverse spectral problems for a new class of nonlocal operators on metric graphs. The Laplace differential operator is considered on a star-shaped graph with nonlocal integral…
Discrete analogs of the Lebedev-Skalskaya transforms are introduced and investigated. It involves series and integrals with respect to the kernels ${\rm Re} K_{\alpha+in}(x), {\rm Im} K_{\alpha+in}(x), x >0, n \in \mathbb{N}, |\alpha | <…
We construct some inverse-closed algebras of bounded integral operators with operator-valued kernels, acting in spaces of vector-valued functions on locally compact groups. To this end we make systematic use of covariance algebras…
In this paper, we propose a new concept of derivative with respect to an arbitrary kernel-function. Several properties related to this new operator, like inversion rules, integration by parts, etc. are studied. In particular, we introduce…
The infinitesimal generator of a one-dimensional strictly $\alpha$-stable process can be represented as a weighted sum of (right and left) Riemann-Liouville fractional derivatives of order $\alpha$ and one obtains the fractional Laplacian…
In this work, we construct an explicit, theoretically rigorous deconvolution method that relies entirely on iterative forward convolutions, thus can be numerically implemented. We first prove that convolution with an even Schwartz kernel…
In this work, we present a compositional safety controller synthesis approach for the class of discrete-time linear control systems. Here, we leverage a state-of-the-art result on the computation of robust controlled invariant sets. To…
This paper investigates the eigenvalue problem of integral operators whose kernels can be expressed as a finite sum of pairwise products of single-variable functions, making them separable. By consdiering the matrix form of the separable…
Lyapunov functions play a vital role in the context of control theory for nonlinear dynamical systems. Besides its classical use for stability analysis, Lyapunov functions also arise in iterative schemes for computing optimal feedback laws…
Much recent work has addressed the solution of a family of partial differential equations by computing the inverse operator map between the input and solution space. Toward this end, we incorporate function-valued reproducing kernel Hilbert…
Inverse problem for Dirac systems with locally square summable potentials and rectangular Weyl functions is solved. For that purpose we use a new result on the linear similarity between operators from a subclass of triangular integral…
Many correct-by-construction control synthesis methods suffer from the curse of dimensionality. Motivated by this challenge, we seek to reduce a correct-by-construction control synthesis problem to subproblems of more modest dimension. As a…
We present a dual form of Lyapunov-Krasovskii functional which allows the problem of controller synthesis of multi-delay systems to be formulated and solved in a convex manner. First, we give a general form of dual stability condition…
For a real irreducible dual pair there is an integral kernel operator which maps the distribution character of an irreducible admissible representation of the group with the smaller or equal rank to an invariant eigendistribution on the…
This note deals with the direct and inverse spectral analysis for a class of infinite band symmetric matrices. This class corresponds to operators arising from difference quations with usual and inner boundary conditions. We give a…
Dunkl operators associated with finite reflection groups generate a commutative algebra of differential-difference operators. There exists a unique linear operator called intertwining operator which intertwines between this algebra and the…
We consider the problem of learning a linear operator $\theta$ between two Hilbert spaces from empirical observations, which we interpret as least squares regression in infinite dimensions. We show that this goal can be reformulated as an…
Operator learning offers a robust framework for approximating mappings between infinite-dimensional function spaces. It has also become a powerful tool for solving inverse problems in the computational sciences. This chapter surveys…