Related papers: Formal proofs of operator identities by a single f…
An invaluable feature of computer algebra systems is their ability to plot the graph of functions. Unfortunately, when one is trying to design a library of mathematical functions, this feature often falls short, producing incorrect and…
The class of absolutely norming operators on complex Hilbert spaces of arbitrary dimensions was introduced in [6] and a spectral characterization theorem for these operators was established in [11]. In this paper we extend the concept of…
We consider the problem of identifying a linear deterministic operator from an input-output measurement. For the large class of continuous (and hence bounded) operators, under additional mild restrictions, we show that stable…
Quantum physics can only make statistical predictions about possible measurement outcomes, and these predictions originate from an operator algebra that is fundamentally different from the conventional definition of probability as a…
We consider kernels of discrete convolution operators or, equivalently, homogeneous solutions of partial difference operators and show that these solutions always have to be exponential polynomials. The respective polynomial space in…
In this paper we develop the formalism of rational complex Bezier curves. This framework is a simple extension of the CAD paradigm, since it describes arc of curves in terms of control polygons and weights, which are extended to complex…
We propose an extension of the framework for discussing the computational complexity of problems involving uncountably many objects, such as real numbers, sets and functions, that can be represented only through approximation. The key idea…
In the framework of (vector valued) quantized holomorphic functions defined on non-commutative spaces, ``quantized hermitian symmetric spaces'', we analyze what the algebras of quantized differential operators with variable coefficients…
The paper aims to provide a full characterization of all operators $T\colon \mathscr{P}(\mathbb{C}) \to \mathscr{P}(\mathbb{C})$ acting on the space of all complex polynomials that satisfy the Leibniz rule \[ T(f\cdot g)= T(f)\cdot g+f\cdot…
The classical Newtonian potentials, defined in terms of metrics, give rise to the basic family of kernels defining linear integral operators and posing the fundamental problems of linear harmonic analysis. When the binary character of a…
Let $\mathbb{K}$ be a field, $\mathcal{X}$ be an infinite set (of indeterminates), and $\mathcal{G}$ be a group acting on $\mathcal{X}$. An ideal in the polynomial ring $\mathbb{K}[\mathcal{X}]$ is called equivariant if it is invariant…
Unlike the classical polynomial case there has not been invented up to very recently a tool similar to the Bernstein-Bezier representation which would allow us to control the behavior of the exponential polynomials. The exponential analog…
The operational calculus associated with special polynomials has proven to be a powerful tool for analyzing and simplifying their properties. This article examines the bivariate degenerate Hermite polynomials with a focus on their…
In this paper we consider several problems of joint similarity to tuples of bounded linear operators in noncommutative polydomains and varieties associated with sets of noncommutative polynomials. We obtain analogues of classical results…
Let us denote ${\cal V}$, the finite dimensional vector spaces of functions of the form $\psi(x) = p_n(x) + f(x) p_m(x)$ where $p_n(x)$ and $p_m(x)$ are arbitrary polynomials of degree at most $n$ and $m$ in the variable $x$ while $f(x)$…
Some properties and relations satisfied by the polynomial solutions of a bispectral problem are studied. Given a finite order differential operator, under certain restrictions, its polynomial eigenfunctions are explicitly obtained, as well…
Given a multivariate real (or complex) polynomial $p$ and a domain $\cal D$, we would like to decide whether an algorithm exists to evaluate $p(x)$ accurately for all $x \in {\cal D}$ using rounded real (or complex) arithmetic. Here…
We formulate several polynomial identities. One side of these identities has a nice simple form. Whereas the other has a form of a polynomial whose coefficients contain binomial coefficients double factorials or (and) rising factorials. The…
Nonlinear dynamical systems can be made easier to control by lifting them into the space of observable functions, where their evolution is described by the linear Koopman operator. This paper describes how the Koopman operator can be used…
In this research, the Bernoulli polynomials are introduced. The properties of these polynomials are employed to construct the operational matrices of integration together with the derivative and product. These properties are then utilized…