Related papers: Pad\'e approximation for a class of hypergeometric…
Several construction methods for rational approximations to functions of one real variable are described in the present paper; the computational results that characterize the comparative accuracy of these methods are presented; an effect of…
A reasonably complete theory of the approximation of an irrational by rational fractions whose numerators and denominators lie in prescribed arithmetic progressions is developed in this paper. Results are both, on the one hand, from a…
We show how the theory of linear forms in two logarithms allows one to get effective irrationality measures for $n$-th roots of rational numbers ${a \over b}$, when $a$ is very close to $b$. We give a $p$-adic analogue of this result under…
We prove new linear independence results for the values of generalized hypergeometric functions ${}_pF_q$ at several distinct algebraic points, over arbitrary algebraic number fields. Our approach combines constructions of type II Pad\'{e}…
In [A. Berele, Computing super matrix invariants, {\it Advances in Applied Math. \bf48} (2012), 273--289.] we defined integrals that approximated the Poincar\'e series of the invariants and concomitants of the general linear Lie supergroup…
We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.
We give necessary and sufficient conditions for the convergence with geometric rate of the denominators of linear Pad\'e-orthogonal approximants corresponding to a measure supported on a general compact set in the complex plane. Thereby, we…
We address the Poincar\'e-Perron's classical problem of approximation for high order linear differential equations in the class of almost periodic type functions, extending the results for a second order linear differential equation in…
This paper is concerned with minimizing a sum of rational functions over a compact set of high-dimension. Our approach relies on the second Lasserre's hierarchy (also known as the upper bounds hierarchy) formulated on the pushforward…
In this paper, we consider the simultaneous approximation of real points by rational points with the error of approximation given by the functions of `non-standard' heights. We prove analogues of Khintchine and Jarn\'ik-Besicovitch theorems…
We find asymptotic equalities for exact upper bounds of approximations by Fourier sums in uniform metric on classes of $2\pi$-periodic functions, representable in the form of convolutions of functions $\varphi$, which belong to unit balls…
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 shall consider a result of Fel'dman, where a sharp Baker-type lower bound is obtained for linear forms in the values of some E-functions. Fel'dman's proof is based on an explicit construction of Pad\'e approximations of the first kind…
Pad\'e approximants to the many-body Green's function can be built by rearranging terms of its perturbative expansion. The hypothesis that the best use of a finite number of terms of such an expansion is given by the subclass of diagonal…
We study the convergence of type I Hermite-Pad\'e approximation for a class of meromorphic functions obtained by adding a vector of rational functions with real coefficients to a Nikishin system of functions.
We prove convergence rates of linear sampling recovery of functions in abstract Bochner spaces satisfying weighted summability of their generalized polynomial chaos expansion coefficients. The underlying algorithm is a function-valued…
A rational approximation by a ratio of polynomial functions is a flexible alternative to polynomial approximation. In particular, rational functions exhibit accurate estimations to nonsmooth and non- Lipschitz functions, where polynomial…
This paper has two purposes. First we present a new definition of the multivariate Pad\'e approximation, a new fast numerical method. Then numerical solution of the one-dimensional (1D) time-dependent nonlinear Sine-Gordon equation (SGE) is…
Operator learning is a recent development in the simulation of Partial Differential Equations (PDEs) by means of neural networks. The idea behind this approach is to learn the behavior of an operator, such that the resulting neural network…
Convergence of diagonal Pad\'e approximants is studied for a class of functions which admit the integral representation $ {\mathfrak F}(\lambda)=r_1(\lambda)\int_{-1}^1\frac{td\sigma(t)}{t-\lambda}+r_2(\lambda), $ where $\sigma$ is a finite…