Related papers: Generic properties of Pad\'e approximants and Pad\…
Starting from the orthogonal polynomial expansion of a function $F$ corresponding to a finite positive Borel measure with infinite compact support, we study the asymptotic behavior of certain associated rational functions…
In the article we propose a general scheme for solutions of some approximation problems under a rather general setting. We illustrate the application of the proposed scheme by a series of examples, in particular we show that many results in…
Recently, a geometrical characterization of vector spaces served to generalize them into a new class of algebras. Instead of the algebraic properties of the underlying fields, we generalized the recently discovered property of such spaces…
In this paper a generalisation of the notion of polarity is exhibited which allows to completely describe, in an incidence-geometric way, the linear complexes of $h$-subspaces. A generalised polarity is defined to be a partial map which…
Given a vector function ${\bf F}=(F_1,\ldots,F_d),$ analytic on a neighborhood of some compact subset $E$ of the complex plane with simply connected complement, we define a sequence of vector rational functions with common denominator in…
Let $\omega_0,\dots,\omega_M$ be complex numbers. If $H_0,\dots,H_M$ are polynomials of degree at most $\rho_0,\dots,\rho_M$, and $G(z)=\sum_{m=0} ^M H_m(z) (1-z)^{\omega_m}$ has a zero at $z=0$ of maximal order (for the given…
We study shallow and deep neural networks whose inputs range over a general topological space. The model is built from a prescribed family of continuous feature maps and reduces to multilayer feedforward networks in the Euclidean case. We…
In this paper, we obtain the analytical solutions of two kinds of transcendental equations with numerous applications in college physics by means of Lagrange inversion theorem, and rewrite them in the form of ratio of rational polynomials…
In this paper we presents further developments regarding the enrichment of the basic Theory of Order Completion. In particular, spaces of generalized functions are constructed that contain generalized solutions to all systems of continuous,…
Truncated perturbative series (TPS's) of any observable have the unphysical dependence on the choice of the renormalization scale (RScl). The diagonal Pad\'e approximants (dPA's) to any TPS of an observable possess the favorable property of…
We present an overview of some recent developments in the theory of generalized formal series, grounded in diffeological geometric framework. These constructions aim to offer new tools for understanding infinite-dimensional phenomena in…
Given a frequency $\lambda$, we study general Dirichlet series $\sum a_n e^{-\lambda_n s}$. First, we give a new condition on $\lambda$ which ensures that a somewhere convergent Dirichlet series defining a bounded holomorphic function in…
Tilings and point sets arising from substitutions are classical mathematical models of quasicrystals. Their hierarchical structure allows one to obtain concrete answers regarding spectral questions tied to the underlying measures and…
We study generalized splines from the perspective of the representation theory of the category of graphs with contractions. Our main theorem proves a kind of finite generation, which in turn implies the existence of a ``universal generating…
We apply the technique of self-similar exponential approximants based on successive truncations of continued exponentials to reconstruct functional laws of the quasi-exponential class from the knowledge of only a few terms of their power…
A representation of the Pad\'e approximation of the $Z$-transform of a signal as a resolvent of a tridiagonal matrix $J$ is given. Several formulas for the poles, zeros and residues of the Pad\'e approximation in terms of the matrix $J$ are…
On a reduced analytic space $X$ we introduce the concept of a generalized cycle, which extends the notion of a formal sum of analytic subspaces to include also a form part. We then consider a suitable equivalence relation and corresponding…
We initiate a study on a range of new generalized derivations of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero. This new generalization of derivations has an analogue in the theory of associative…
The primary purpose of this paper concerns the relation of (compact) generalized manifolds to finite Poincar\'{e} duality complexes (PD complexes). The problem is that an arbitrary generalized manifold $X$ is always an ENR space, but it is…
The recent significant enrichment of the Order Completion Method for nonlinear Systems of PDEs resulted in the global existence of generalized solutions to a large class of such equations. In this paper we investigate the existence and…