Estimates for solutions of systems of linear equations with circulant matrices

In the present paper we consider the problem to estimate a solution of the system of equations with a circulant matrix in uniform norm. We give the estimate for circulant matrices with diagonal dominance. The estimate is sharp. Based on this result and an idea of decomposition of the matrix into a product of matrices associated with factorization of the characteristic polynomial, we propose an estimate for any circulant matrix.

Talagrand’s quadratic transportation cost inequalities for SPDEs driven by fractional noises with two reflection walls

Using the method of Girsanov’s transformation, we investigate Talagrand’s quadratic transportation cost inequalities for the laws of the solutions of stochastic partial differential equations (SPDEs) with two reflection walls under the uniform norm on the continuous functions space. These equations are driven by fractional noises.

Banach-Steinhaus Theorem for the Space P of All Primitives of Henstock-Kurzweil Integrable Functions

In this paper, it is shown how the Banach-Steinhaus theorem for the space P of all primitives of Henstock-Kurzweil integrable functions on a closed bounded interval, equipped with the uniform norm, can follow from the Banach-Steinhaus theorem for the Denjoy space by applying the classical Hahn-Banach theorem and Riesz representation theorem.

On Properties of a Regular Simplex Inscribed into a Ball

Let $B$ be a Euclidean ball in ${\mathbb R}^n$ and let $C(B)$ be a space of continuos functions $f:B\to{\mathbb R}$ with the uniform norm $\|f\|_{C(B)}:=\max_{x\in B}|f(x)|.$ By $\Pi_1\left({\mathbb R}^n\right)$ we mean a set of polynomials of degree $\leq 1$, i.e., a set of linear functions upon ${\mathbb R}^n$. The interpolation projector $P:C(B)\to \Pi_1({\mathbb R}^n)$ with the nodes $x^{(j)}\in B$ is defined by the equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right)$, $j=1,\ldots, n+1$.The norm of $P$ as an operator from $C(B)$ to $C(B)$ can be calculated by the formula $\|P\|_B=\max_{x\in B}\sum |\lambda_j(x)|.$ Here $\lambda_j$ are the basic Lagrange polynomials corresponding to the $n$-dimensional nondegenerate simplex $S$ with the vertices $x^{(j)}$. Let $P^\prime$ be a projector having the nodes in the vertices of a regular simplex inscribed into the ball. We describe the points $y\in B$ with the property $\|P^\prime\|_B=\sum |\lambda_j(y)|$. Also we formulate some geometric conjecture which implies that $\|P^\prime\|_B$ is equal to the minimal norm of an interpolation projector with nodes in $B$. We prove that this conjecture holds true at least for $n=1,2,3,4$.

Neural Network Identifiability for a Family of Sigmoidal Nonlinearities

This paper addresses the following question of neural network identifiability: Does the input–output map realized by a feed-forward neural network with respect to a given nonlinearity uniquely specify the network architecture, weights, and biases? The existing literature on the subject (Sussman in Neural Netw 5(4):589–593, 1992; Albertini et al. in Artificial neural networks for speech and vision, 1993; Fefferman in Rev Mat Iberoam 10(3):507–555, 1994) suggests that the answer should be yes, up to certain symmetries induced by the nonlinearity, and provided that the networks under consideration satisfy certain “genericity conditions.” The results in Sussman (1992) and Albertini et al. (1993) apply to networks with a single hidden layer and in Fefferman (1994) the networks need to be fully connected. In an effort to answer the identifiability question in greater generality, we derive necessary genericity conditions for the identifiability of neural networks of arbitrary depth and connectivity with an arbitrary nonlinearity. Moreover, we construct a family of nonlinearities for which these genericity conditions are minimal, i.e., both necessary and sufficient. This family is large enough to approximate many commonly encountered nonlinearities to within arbitrary precision in the uniform norm.

Estimate of derivatives of functions on the real line in Weil spaces

We prove the inequality that estimates seminorm of Weil of the derivatives of the functions on the real line with the help of uniform norm of the functions and their derivatives. We also solve the corresponding problem of Kolmogorov.

Quantitative Stability in the Geometry of Semi-discrete Optimal Transport

We show quantitative stability results for the geometric “cells” arising in semi-discrete optimal transport problems. We first show stability of the associated Laguerre cells in measure, without any connectedness or regularity assumptions on the source measure. Next we show quantitative invertibility of the map taking dual variables to the measures of Laguerre cells, under a Poincarè-Wirtinger inequality. Combined with a regularity assumption equivalent to the Ma–Trudinger–Wang conditions of regularity in Monge-Ampère, this invertibility leads to stability of Laguerre cells in Hausdorff measure and also stability in the uniform norm of the dual potential functions, all stability results come with explicit quantitative bounds. Our methods utilize a combination of graph theory, convex geometry, and Monge-Ampère regularity theory.

INEQUALITIES FOR ALGEBRAIC POLYNOMIALS ON AN ELLIPSE

The paper presents new solutions to two classical problems of approximation theory. The first problem is to find the polynomial that deviates least from zero on an ellipse. The second one is to find the exact upper bound of the uniform norm on an ellipse with foci \(\pm 1\) of the derivative of an algebraic polynomial with real coefficients normalized on the segment \([- 1,1]\).